Let f(x) = { x^2, x is irrational ; x , x is rational. A) Discuss the continuity of f . B) Discuss the differentiability of f . C) Does the integral from 0 to 1 of f(x) is exist?

Answer 1

This function is continuous and differentiable for x=0 and x=1. This function is not Riemann integrable on the given interval, since a function must be continuous on the entire interval. However, it is Lebesgue integable, but this is probably too advanced for now.