Forgive my lack of syntax here

if f is continuous at c, then for e> 0 there exists a d>0 st

0< l x - c l < d --> l f(x) - f(c) l < e

but l f(x) - f(c) l < e is the same thing as:

f(c) - e < f(x) < f(c) + e

so as long as 0 < e < f(c) we have f(x) > 0 for all 0 < l x - c l < d

and the conclusion follows