derivative of function and fundamental theorem of calculus
Let $fcolon[a,b]tomathbb{R}$ be differentiable on $(a,b)$. Suppose that the limits $f(a+)=lim_{xto a+}f(x)$ and $f(b-)=lim_{xto b-}f(x)$ exist and are finite.
My question is: Do we have
$$int_{a}^{b}f'(x)dx=f(b-)-f(a+)$$
without further assumption on $f$? If yes, what would be a reference for this result? If no, is there a counterexample for this?
Any help is highly appreciated.
calculus real-analysis integration derivatives
asked Nov 28 '18 at 18:26
weirdo
420210
add a comment |
Let $fcolon[a,b]tomathbb{R}$ be differentiable on $(a,b)$. Suppose that the limits $f(a+)=lim_{xto a+}f(x)$ and $f(b-)=lim_{xto b-}f(x)$ exist and are finite.
My question is: Do we have
$$int_{a}^{b}f'(x)dx=f(b-)-f(a+)$$
without further assumption on $f$? If yes, what would be a reference for this result? If no, is there a counterexample for this?
Any help is highly appreciated.
calculus real-analysis integration derivatives
asked Nov 28 '18 at 18:26
weirdo
420210
add a comment |
Let $fcolon[a,b]tomathbb{R}$ be differentiable on $(a,b)$. Suppose that the limits $f(a+)=lim_{xto a+}f(x)$ and $f(b-)=lim_{xto b-}f(x)$ exist and are finite.
My question is: Do we have
$$int_{a}^{b}f'(x)dx=f(b-)-f(a+)$$
without further assumption on $f$? If yes, what would be a reference for this result? If no, is there a counterexample for this?
Any help is highly appreciated.
calculus real-analysis integration derivatives
asked Nov 28 '18 at 18:26
weirdo
420210
Let $fcolon[a,b]tomathbb{R}$ be differentiable on $(a,b)$. Suppose that the limits $f(a+)=lim_{xto a+}f(x)$ and $f(b-)=lim_{xto b-}f(x)$ exist and are finite.
My question is: Do we have
$$int_{a}^{b}f'(x)dx=f(b-)-f(a+)$$
without further assumption on $f$? If yes, what would be a reference for this result? If no, is there a counterexample for this?
Any help is highly appreciated.
calculus real-analysis integration derivatives
calculus real-analysis integration derivatives
asked Nov 28 '18 at 18:26
weirdo
420210
asked Nov 28 '18 at 18:26
weirdo
420210
edited Nov 28 '18 at 18:36
asked Nov 28 '18 at 18:26
weirdo
420210
asked Nov 28 '18 at 18:26
weirdo
420210
asked Nov 28 '18 at 18:26
weirdo
420210
420210
add a comment |
add a comment |
1 Answer
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No, this is not true. In fact, it's not even necessarily true that $f'$ is integrable. The classical example of such a pathological counterexample is Volterra's function.
answered Nov 28 '18 at 19:13
davidlowryduda♦
74.4k7117251
add a comment |
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No, this is not true. In fact, it's not even necessarily true that $f'$ is integrable. The classical example of such a pathological counterexample is Volterra's function.
answered Nov 28 '18 at 19:13
davidlowryduda♦
74.4k7117251
add a comment |
No, this is not true. In fact, it's not even necessarily true that $f'$ is integrable. The classical example of such a pathological counterexample is Volterra's function.
answered Nov 28 '18 at 19:13
davidlowryduda♦
74.4k7117251
add a comment |
No, this is not true. In fact, it's not even necessarily true that $f'$ is integrable. The classical example of such a pathological counterexample is Volterra's function.
answered Nov 28 '18 at 19:13
davidlowryduda♦
74.4k7117251
No, this is not true. In fact, it's not even necessarily true that $f'$ is integrable. The classical example of such a pathological counterexample is Volterra's function.
answered Nov 28 '18 at 19:13
davidlowryduda♦
74.4k7117251
answered Nov 28 '18 at 19:13
davidlowryduda♦
74.4k7117251
answered Nov 28 '18 at 19:13
davidlowryduda♦
74.4k7117251
answered Nov 28 '18 at 19:13
davidlowryduda♦
74.4k7117251
74.4k7117251
add a comment |
add a comment |
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