Pointwise convergence but not uniformly convergence of $g_{n} to 0$ when $g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+...











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Let $f:[0,1] to mathbb{R}$ a continuous function such $f(0)=f(frac{1}{2})=0$ and $(f[0, frac{1}{2}]) not subset lbrace 0 rbrace$. For each $n in mathbb{N}$, let $g_{n}:[0, infty ) to mathbb{R}$ defined for each $x in [0, infty)$ as $$g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+ sqrt[n]{x}}right)$$
Prove the following:



(a) $lbrace g_{n} rbrace_{n=1}^{infty}$ is pointwise convergent to $0$.



(b) $lbrace g_{n} rbrace_{n=1}^{infty}$ is not uniformly continuous to $0$.



My attemp for (a) goes as follow: As



$$left| frac{sqrt[n]{x}}{1+ sqrt[n]{x}} right|= left| left( sqrt[n]{x}right) frac{1}{1+ sqrt[n]{x}} right| = left| sqrt[n]{x} right| left| frac{1}{1+ sqrt[n]{x}} right|$$. I want to use the fact that as $sqrt[n]{x} to 1$ as $n to infty$ for $x in [0,1]$ and $frac{1}{1+ sqrt[n]{x}} to frac{1}{2}$ as $n to infty$ for $x in [0,1]$ then we can bound



$$left| frac{sqrt[n]{x}}{1+ sqrt[n]{x}} - frac{1}{2} right|< epsilon.$$



For every $epsilon>0$ and of course a particular $ngeq N(epsilon) in mathbb{N}$. Then as $f$ is continuous in [0,1] then



$$ g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+ sqrt[n]{x}}right) to fleft(frac{1}{2}right)=0$$



Im kind of lost in proving the not uniformly continuity of $g_{n} to 0$. Any help ending the proof would be appreciated. Thanks










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    Let $f:[0,1] to mathbb{R}$ a continuous function such $f(0)=f(frac{1}{2})=0$ and $(f[0, frac{1}{2}]) not subset lbrace 0 rbrace$. For each $n in mathbb{N}$, let $g_{n}:[0, infty ) to mathbb{R}$ defined for each $x in [0, infty)$ as $$g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+ sqrt[n]{x}}right)$$
    Prove the following:



    (a) $lbrace g_{n} rbrace_{n=1}^{infty}$ is pointwise convergent to $0$.



    (b) $lbrace g_{n} rbrace_{n=1}^{infty}$ is not uniformly continuous to $0$.



    My attemp for (a) goes as follow: As



    $$left| frac{sqrt[n]{x}}{1+ sqrt[n]{x}} right|= left| left( sqrt[n]{x}right) frac{1}{1+ sqrt[n]{x}} right| = left| sqrt[n]{x} right| left| frac{1}{1+ sqrt[n]{x}} right|$$. I want to use the fact that as $sqrt[n]{x} to 1$ as $n to infty$ for $x in [0,1]$ and $frac{1}{1+ sqrt[n]{x}} to frac{1}{2}$ as $n to infty$ for $x in [0,1]$ then we can bound



    $$left| frac{sqrt[n]{x}}{1+ sqrt[n]{x}} - frac{1}{2} right|< epsilon.$$



    For every $epsilon>0$ and of course a particular $ngeq N(epsilon) in mathbb{N}$. Then as $f$ is continuous in [0,1] then



    $$ g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+ sqrt[n]{x}}right) to fleft(frac{1}{2}right)=0$$



    Im kind of lost in proving the not uniformly continuity of $g_{n} to 0$. Any help ending the proof would be appreciated. Thanks










    share|cite|improve this question


























      up vote
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      down vote

      favorite









      up vote
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      down vote

      favorite











      Let $f:[0,1] to mathbb{R}$ a continuous function such $f(0)=f(frac{1}{2})=0$ and $(f[0, frac{1}{2}]) not subset lbrace 0 rbrace$. For each $n in mathbb{N}$, let $g_{n}:[0, infty ) to mathbb{R}$ defined for each $x in [0, infty)$ as $$g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+ sqrt[n]{x}}right)$$
      Prove the following:



      (a) $lbrace g_{n} rbrace_{n=1}^{infty}$ is pointwise convergent to $0$.



      (b) $lbrace g_{n} rbrace_{n=1}^{infty}$ is not uniformly continuous to $0$.



      My attemp for (a) goes as follow: As



      $$left| frac{sqrt[n]{x}}{1+ sqrt[n]{x}} right|= left| left( sqrt[n]{x}right) frac{1}{1+ sqrt[n]{x}} right| = left| sqrt[n]{x} right| left| frac{1}{1+ sqrt[n]{x}} right|$$. I want to use the fact that as $sqrt[n]{x} to 1$ as $n to infty$ for $x in [0,1]$ and $frac{1}{1+ sqrt[n]{x}} to frac{1}{2}$ as $n to infty$ for $x in [0,1]$ then we can bound



      $$left| frac{sqrt[n]{x}}{1+ sqrt[n]{x}} - frac{1}{2} right|< epsilon.$$



      For every $epsilon>0$ and of course a particular $ngeq N(epsilon) in mathbb{N}$. Then as $f$ is continuous in [0,1] then



      $$ g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+ sqrt[n]{x}}right) to fleft(frac{1}{2}right)=0$$



      Im kind of lost in proving the not uniformly continuity of $g_{n} to 0$. Any help ending the proof would be appreciated. Thanks










      share|cite|improve this question















      Let $f:[0,1] to mathbb{R}$ a continuous function such $f(0)=f(frac{1}{2})=0$ and $(f[0, frac{1}{2}]) not subset lbrace 0 rbrace$. For each $n in mathbb{N}$, let $g_{n}:[0, infty ) to mathbb{R}$ defined for each $x in [0, infty)$ as $$g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+ sqrt[n]{x}}right)$$
      Prove the following:



      (a) $lbrace g_{n} rbrace_{n=1}^{infty}$ is pointwise convergent to $0$.



      (b) $lbrace g_{n} rbrace_{n=1}^{infty}$ is not uniformly continuous to $0$.



      My attemp for (a) goes as follow: As



      $$left| frac{sqrt[n]{x}}{1+ sqrt[n]{x}} right|= left| left( sqrt[n]{x}right) frac{1}{1+ sqrt[n]{x}} right| = left| sqrt[n]{x} right| left| frac{1}{1+ sqrt[n]{x}} right|$$. I want to use the fact that as $sqrt[n]{x} to 1$ as $n to infty$ for $x in [0,1]$ and $frac{1}{1+ sqrt[n]{x}} to frac{1}{2}$ as $n to infty$ for $x in [0,1]$ then we can bound



      $$left| frac{sqrt[n]{x}}{1+ sqrt[n]{x}} - frac{1}{2} right|< epsilon.$$



      For every $epsilon>0$ and of course a particular $ngeq N(epsilon) in mathbb{N}$. Then as $f$ is continuous in [0,1] then



      $$ g_{n}(x)=fleft(frac{sqrt[n]{x}}{1+ sqrt[n]{x}}right) to fleft(frac{1}{2}right)=0$$



      Im kind of lost in proving the not uniformly continuity of $g_{n} to 0$. Any help ending the proof would be appreciated. Thanks







      calculus real-analysis sequences-and-series complex-analysis continuity






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      edited Nov 17 at 21:35









      rtybase

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      asked Nov 17 at 18:36









      Cos

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          For (a) you need to distinguish between $x=0$ and $x>0.$ If $x=0,$ then $g_n(x) = f(0)=0$ for all $n.$ If $x>0,$ then $x^{1/n} to 1,$ hence $x^{1/n}/(1+x^{1/n}) to 1/2,$ which implies $g_n(x) to f(1/2)=0.$



          For (b), we know that there is $cin (0,1/2)$ such that $f(c)ne 0.$ This $c$ can be written as $x/(1+x)$ for some $xge 0.$ We then have $g_n(x^n) = f(x/(1+x))=f(c)$ for all $n.$ Thus



          $$sup_{mathbb [0,infty)}|g_n| ge |g_n(x^n)| = |f(c)| >0$$



          for all $n.$ This shows $g_n$ does not converge uniformly to $0$ on $[0,infty).$






          share|cite|improve this answer





















          • Thanks! @zhw But I still dont figure out how the last inequality involving the supreme of $g_{n}$ proves $g_{n}$ is not uniformly convergent to zero in the given interval? :(
            – Cos
            Nov 17 at 19:42










          • Take $epsilon= |f(c)|/2.$
            – zhw.
            Nov 17 at 21:46


















          up vote
          0
          down vote













          We show that the function is not uniformly continuous even over $(1,infty)$. First notice that since $f(x)$ is continuous over $[0,1]$ we have:$$0<x<delta_epsilonquad,quad |f(x)|<epsilon $$therefore $$forall n>Nquadto quad 0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilonto |g_n(x)|=|f({sqrt[n]xover 1+sqrt[n]x})|<epsilon$$also $0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilon$ is equivalent to $$n>left({1over delta_epsilon}-1right){1over ln x}=N$$. Uniform continuity requires that $N=left({1over delta_epsilon}-1right){1over ln x}$ be a function only of $epsilon$ while we see that a term $1overln x$ makes it dependent also to $x$. Therefore $g_n(x)$ tends to $0$ though not uniformly.






          share|cite|improve this answer





















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            For (a) you need to distinguish between $x=0$ and $x>0.$ If $x=0,$ then $g_n(x) = f(0)=0$ for all $n.$ If $x>0,$ then $x^{1/n} to 1,$ hence $x^{1/n}/(1+x^{1/n}) to 1/2,$ which implies $g_n(x) to f(1/2)=0.$



            For (b), we know that there is $cin (0,1/2)$ such that $f(c)ne 0.$ This $c$ can be written as $x/(1+x)$ for some $xge 0.$ We then have $g_n(x^n) = f(x/(1+x))=f(c)$ for all $n.$ Thus



            $$sup_{mathbb [0,infty)}|g_n| ge |g_n(x^n)| = |f(c)| >0$$



            for all $n.$ This shows $g_n$ does not converge uniformly to $0$ on $[0,infty).$






            share|cite|improve this answer





















            • Thanks! @zhw But I still dont figure out how the last inequality involving the supreme of $g_{n}$ proves $g_{n}$ is not uniformly convergent to zero in the given interval? :(
              – Cos
              Nov 17 at 19:42










            • Take $epsilon= |f(c)|/2.$
              – zhw.
              Nov 17 at 21:46















            up vote
            0
            down vote













            For (a) you need to distinguish between $x=0$ and $x>0.$ If $x=0,$ then $g_n(x) = f(0)=0$ for all $n.$ If $x>0,$ then $x^{1/n} to 1,$ hence $x^{1/n}/(1+x^{1/n}) to 1/2,$ which implies $g_n(x) to f(1/2)=0.$



            For (b), we know that there is $cin (0,1/2)$ such that $f(c)ne 0.$ This $c$ can be written as $x/(1+x)$ for some $xge 0.$ We then have $g_n(x^n) = f(x/(1+x))=f(c)$ for all $n.$ Thus



            $$sup_{mathbb [0,infty)}|g_n| ge |g_n(x^n)| = |f(c)| >0$$



            for all $n.$ This shows $g_n$ does not converge uniformly to $0$ on $[0,infty).$






            share|cite|improve this answer





















            • Thanks! @zhw But I still dont figure out how the last inequality involving the supreme of $g_{n}$ proves $g_{n}$ is not uniformly convergent to zero in the given interval? :(
              – Cos
              Nov 17 at 19:42










            • Take $epsilon= |f(c)|/2.$
              – zhw.
              Nov 17 at 21:46













            up vote
            0
            down vote










            up vote
            0
            down vote









            For (a) you need to distinguish between $x=0$ and $x>0.$ If $x=0,$ then $g_n(x) = f(0)=0$ for all $n.$ If $x>0,$ then $x^{1/n} to 1,$ hence $x^{1/n}/(1+x^{1/n}) to 1/2,$ which implies $g_n(x) to f(1/2)=0.$



            For (b), we know that there is $cin (0,1/2)$ such that $f(c)ne 0.$ This $c$ can be written as $x/(1+x)$ for some $xge 0.$ We then have $g_n(x^n) = f(x/(1+x))=f(c)$ for all $n.$ Thus



            $$sup_{mathbb [0,infty)}|g_n| ge |g_n(x^n)| = |f(c)| >0$$



            for all $n.$ This shows $g_n$ does not converge uniformly to $0$ on $[0,infty).$






            share|cite|improve this answer












            For (a) you need to distinguish between $x=0$ and $x>0.$ If $x=0,$ then $g_n(x) = f(0)=0$ for all $n.$ If $x>0,$ then $x^{1/n} to 1,$ hence $x^{1/n}/(1+x^{1/n}) to 1/2,$ which implies $g_n(x) to f(1/2)=0.$



            For (b), we know that there is $cin (0,1/2)$ such that $f(c)ne 0.$ This $c$ can be written as $x/(1+x)$ for some $xge 0.$ We then have $g_n(x^n) = f(x/(1+x))=f(c)$ for all $n.$ Thus



            $$sup_{mathbb [0,infty)}|g_n| ge |g_n(x^n)| = |f(c)| >0$$



            for all $n.$ This shows $g_n$ does not converge uniformly to $0$ on $[0,infty).$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 17 at 18:56









            zhw.

            70.5k43075




            70.5k43075












            • Thanks! @zhw But I still dont figure out how the last inequality involving the supreme of $g_{n}$ proves $g_{n}$ is not uniformly convergent to zero in the given interval? :(
              – Cos
              Nov 17 at 19:42










            • Take $epsilon= |f(c)|/2.$
              – zhw.
              Nov 17 at 21:46


















            • Thanks! @zhw But I still dont figure out how the last inequality involving the supreme of $g_{n}$ proves $g_{n}$ is not uniformly convergent to zero in the given interval? :(
              – Cos
              Nov 17 at 19:42










            • Take $epsilon= |f(c)|/2.$
              – zhw.
              Nov 17 at 21:46
















            Thanks! @zhw But I still dont figure out how the last inequality involving the supreme of $g_{n}$ proves $g_{n}$ is not uniformly convergent to zero in the given interval? :(
            – Cos
            Nov 17 at 19:42




            Thanks! @zhw But I still dont figure out how the last inequality involving the supreme of $g_{n}$ proves $g_{n}$ is not uniformly convergent to zero in the given interval? :(
            – Cos
            Nov 17 at 19:42












            Take $epsilon= |f(c)|/2.$
            – zhw.
            Nov 17 at 21:46




            Take $epsilon= |f(c)|/2.$
            – zhw.
            Nov 17 at 21:46










            up vote
            0
            down vote













            We show that the function is not uniformly continuous even over $(1,infty)$. First notice that since $f(x)$ is continuous over $[0,1]$ we have:$$0<x<delta_epsilonquad,quad |f(x)|<epsilon $$therefore $$forall n>Nquadto quad 0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilonto |g_n(x)|=|f({sqrt[n]xover 1+sqrt[n]x})|<epsilon$$also $0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilon$ is equivalent to $$n>left({1over delta_epsilon}-1right){1over ln x}=N$$. Uniform continuity requires that $N=left({1over delta_epsilon}-1right){1over ln x}$ be a function only of $epsilon$ while we see that a term $1overln x$ makes it dependent also to $x$. Therefore $g_n(x)$ tends to $0$ though not uniformly.






            share|cite|improve this answer

























              up vote
              0
              down vote













              We show that the function is not uniformly continuous even over $(1,infty)$. First notice that since $f(x)$ is continuous over $[0,1]$ we have:$$0<x<delta_epsilonquad,quad |f(x)|<epsilon $$therefore $$forall n>Nquadto quad 0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilonto |g_n(x)|=|f({sqrt[n]xover 1+sqrt[n]x})|<epsilon$$also $0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilon$ is equivalent to $$n>left({1over delta_epsilon}-1right){1over ln x}=N$$. Uniform continuity requires that $N=left({1over delta_epsilon}-1right){1over ln x}$ be a function only of $epsilon$ while we see that a term $1overln x$ makes it dependent also to $x$. Therefore $g_n(x)$ tends to $0$ though not uniformly.






              share|cite|improve this answer























                up vote
                0
                down vote










                up vote
                0
                down vote









                We show that the function is not uniformly continuous even over $(1,infty)$. First notice that since $f(x)$ is continuous over $[0,1]$ we have:$$0<x<delta_epsilonquad,quad |f(x)|<epsilon $$therefore $$forall n>Nquadto quad 0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilonto |g_n(x)|=|f({sqrt[n]xover 1+sqrt[n]x})|<epsilon$$also $0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilon$ is equivalent to $$n>left({1over delta_epsilon}-1right){1over ln x}=N$$. Uniform continuity requires that $N=left({1over delta_epsilon}-1right){1over ln x}$ be a function only of $epsilon$ while we see that a term $1overln x$ makes it dependent also to $x$. Therefore $g_n(x)$ tends to $0$ though not uniformly.






                share|cite|improve this answer












                We show that the function is not uniformly continuous even over $(1,infty)$. First notice that since $f(x)$ is continuous over $[0,1]$ we have:$$0<x<delta_epsilonquad,quad |f(x)|<epsilon $$therefore $$forall n>Nquadto quad 0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilonto |g_n(x)|=|f({sqrt[n]xover 1+sqrt[n]x})|<epsilon$$also $0<{sqrt[n]xover 1+sqrt[n]x}<delta_epsilon$ is equivalent to $$n>left({1over delta_epsilon}-1right){1over ln x}=N$$. Uniform continuity requires that $N=left({1over delta_epsilon}-1right){1over ln x}$ be a function only of $epsilon$ while we see that a term $1overln x$ makes it dependent also to $x$. Therefore $g_n(x)$ tends to $0$ though not uniformly.







                share|cite|improve this answer












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                share|cite|improve this answer










                answered 2 days ago









                Mostafa Ayaz

                13k3735




                13k3735






























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