Exercise 7.5.4

Question

Show that if $f:[a, b] \rightarrow \mathbf{R}$ is continuous and $\int_{a}^{x} f=0$ for all $x \in[a, b]$, then $f(x)=0$ everywhere on $[a, b]$. Provide an example to show that this conclusion does not follow if $f$ is not continuous.

Ulisse Mini · Accepted Answer

Since $f$ is continuous, by Theorem 7.5.1 (ii), letting $F(x) = \int^x_a f = 0$, we have $f(x) = F'(x) = 0$. If $f$ is not continuous, then this does not hold (e.g. Thomae's function over $[0,1]$)

Exercise 7.5.4

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Exercise 7.5.4

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