Discuss the continuity, differentiability, integrability of the function

Question

Discuss the continuity, differentiability, integrability of the function $$f(x)=\begin{cases} x^2,x\in \mathbb{Q} \\ 0, x\in \mathbb{R/Q}\end{cases}$$ Is the function discontinuous at everywhere? I got this by discussing if the point is rational or irrational. And construct a rational sequence converges to irrational point , an irrational sequence to rational point.
Now if the function is discontinuous then we can easily conclude it is not differentiable.
And I don't know how to discuss the integrability of the function.

Answer 1

Hints

• The function is continuous at $0$ because $|f(x)|\le x^2.$
• The function is differentiable at $0$ because the limit $\lim_{x\to 0}\dfrac{f(x)-f(0)}{x}$ exits.
• Assuming you are asking about Riemann integrability: What can you say about the set about the set of points of discontinuity? Does it have any relation with being integrable?