Give an example of Riemann Integrable function which has a primitive. Justify

Question

Give an example of Riemann Integrable function (discontinuous) which has a primitive. Justify

Please help me. I need an example of such a discontinuous function.

Is $f(x)=\cos{1/x}$ a right choice

Answer 1

Consider $$f(x) = \begin{cases} x^2 \sin(x^{-1}) & \text{if } x \neq 0 \\ 0 & \text{if }x = 0 \end{cases}.$$ Then $$f'(x) = \begin{cases} 2x \sin(x^{-1}) - \cos(x^{-1}) & \text{if } x \neq 0 \\ 0 & \text{if }x = 0 \end{cases}.$$ (The derivative at $0$ follows from the derivative squeeze theorem, as $-x^2 \le f(x) \le x^2$.) Note that $f'$ is discontinuous at $0$, and has a primitive $f$.

All that remains is showing that $f'$ is Riemann Integrable. Note that we may express

$$f'(x) = \begin{cases} 2x \sin(x^{-1}) & \text{if } x \neq 0 \\ 0 & \text{if }x = 0 \end{cases} - \begin{cases} \cos(x^{-1}) & \text{if } x \neq 0 \\ 0 & \text{if }x = 0 \end{cases}.$$

The former terms is continuous by the squeeze theorem, and hence is integrable over any compact interval. The latter term can be proven Riemann Integrable by modifying my answer here.