If if f is integrable on [a, b] show by example that F(t)= integral from a to t f( x ) dx is continuous but need not to be differentiable

Question

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

F(t)= integral from a to t f( x ) dx

is continuous but need not to be differentiable

Answer 1

Here is an example, on the interval [-1,1] for the function f(x)=-|x|+1.

It is easily shown to be integratable, it is simply an isosceles triangle with height 1, and base length 2. We can find the area of this using simple triangle formulae, and its obviously continuous--nowhere to jump.

This is non-differentiable due to the point at (0,1) being a sharp turn, with perfect 90 degrees. Anytime a function makes a sharp turn (think, if you can find a 'vertex' or 'corner' on a graph, and it is not curved, then this fulfills it) it is non differentiable, as there is no rate of change from one number to another, just a jump.

Remember, anything you can find the area of (and is a function), is integratable, but it is only differentiable if it is smooth.