## 7.1. Riemann Integral

### Examples 7.1.21(c):

Show that if one starts with an integrable function

Our Fundamental Theorem states that if we start with a continuous
function *f*in the Fundamental Theorem of Calculus that is not continuous, the corresponding function*F*may not be differentiable.*f(t)*over some interval

*[a, b]*then the new function

*F(x)*obtained by integrating

*f*from

*a*to some variable value

*x*is differentiable:

Now let's start with a simple step function that is integrable but not continuous over the interval, say,F(x) = f(t) dtis differentiable as long asfis continuous

*[-1, 1]*. Define

Then for if

*x < 0*we have:

and for

*x 0*

But then

*x = 0*.