CALCULUS QUESTION HEEELP ASAP!!!

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

An example can be based on letting f(x) be the step function.

That is f(x) = 0 x < 0

f(0) = 0

f(x) = 1 x > 0

Then (assuming a < 0)

F(x) = 0 x < 0

F(0) = 0

F(x) = x x >0

This is continuous because the rules agree at the junction point, But not differentiable because the slope changes at the junction point.