Show example where $F(x) = \int_0^x f(t) \, dt$ but $F'(x_0) \neq f(x_0)$

Question

Q: Give an example of a Riemann integrable function $f$ and a point $x_0$ so that

\begin{align*} F(x) &= \int_0^x f(t) \, dt \\ \end{align*}

is defined but $F'(x_0) \neq f(x_0)$.

Answer 1

One of the versions of the fundamental theorem of calculus says that if $f$ is a continuous function from a closed interval $[a,b]$ to $\mathbb{R}$ then for any $x \in (a,b)$ we have $$F’(x)=f(x)$$ where $F$ is defined by $F(t)=\int_a^t f(x)dx$ for $t$ in $[a,b]$. Moreover, if an integrable function differs with another function in just one point, the other function is integrable as well with equal integral. So for an example of what you ask for, just take any continuous function on $[a,b]$ and make another function out of it by simply changing the original function at one point in $(a,b)$. So an example would be $f:[0,1] \rightarrow \mathbb{R}$ with $$f(x)=0$$ if $x \neq \frac{1}{2}$ and $f(\frac{1}{2})=1$. Then $F’=0$ but $f$ is not 0 on $(0,1)$.