Secant line in Fermat's Theorem

I'm trying to understand something in Fermat's Theorem. I can't really phrase it in words, but I will write what my textbook says.

Apparently if

$$\lim_{x→c}\frac{f(x)-f(c)}{x-c} > 0$$

then there exists an open interval (a,b) containing c such that

$$\frac{f(x)-f(c)}{x-c} > 0$$ for all c in that interval.

How does this follow from the definition of the derivative?

I appreciate all help.

Thanks!
 Recognitions: Gold Member Homework Help That is a general property of limits. If$$\lim_{x\rightarrow c}g(x) = L > 0$$then there is an open interval ##I## containing ##c## on which ##g(x)>0##. It comes directly from the ##\epsilon - \delta## definition of limit.