HARD CALCULUS QUESTION, HEEELP!!!

Question

If f is integrable on [a, b], show by example that :

t

F (t) = ∫ f(x) dx

a

is continuous but need not to be differentiable.

note: the integral from (a) to (t).

Answer 1

Think about the problem conceptually: Try to think of a simple function f(x) (in the integrand) that is continuous and increasing or decreasing on some interval, but then immediately drops to zero at some point (when its slope is nonzero; note that f(x) does not need to be continuous). Then, when you integrate this, F(x) will be continuous, but will also have a corner. So, it will not be differentiable.

Make sense?