Expert: Paul Klarreich - 1/13/2010

Question
I need a proof on this theorem. I would gladly appreciate if you can help me:  
If f is continuous at R, then F is continuous at each y on the interval [c,d].

Answer
You want to prove:

If  f(x) is continuous on [a,b] and F(x) is an antiderivative of f, then F is continuous on [a,b].

--- I will use a,b as the endpoints, and c as a point in between.

Basically, you want to use the fact that differentiability implies continuity.

To prove:

lim[x-c] F(x) = F(c)

Now  F'(c) = f(c), by definition of F,

            F(x) - F(c)  
So lim[x->c] ------------ = f(c)
               x - c

Since f(c) exists, [part of the def of continuity], and lim[x->c](x - c) = 0,
we must have

lim[x->c]  F(x) - F(c)  = 0

So lim[x->c]  F(x) = F(c).

Next time, be more careful with your notation and you will get an answer more quickly.