Originally Posted by

**ThePerfectHacker** Okay, you have the function

$\displaystyle F(x)=\int_x^a f(t) dt$

It exists since all continous functions are integrable.

Now, what ails me is that the upper limit is $\displaystyle a$, I think you wanted to write $\displaystyle c\in [a,b]$ thus hence the function,

$\displaystyle F(x)=\int_x^c f(t)dt=-\int_c^x f(t)dt$

The derivative of this function is (2nd fun. the. of cal.)

$\displaystyle F'(x)=-f(x)<0$

Thus the function is decreasing NOT increasing.

Thus, I think you wanted to write,

$\displaystyle F(x)=\int_c^x f(t)dt$

Then using the above concepts it is increasing on $\displaystyle [a,b]$