Functions and continuity. Let $f : \mathbb{R} \to \mathbb{R}$ be continuous at $c$ and let $f(c) > 0$. Show that there exists an open interval $I$ containing $c$ such that $f(x) > 0$ for all $x \in I$.

Hint. Recall that continuity implies that $\lim_{x\to c}f(x)=f(c)$. 
Then by the definition of limit,
$$\forall \epsilon>0,\exists\delta>0: |x-c|<\delta \implies |f(x)-f(c)|<\epsilon.$$
This means that, for all $x\in I:=(c-\delta,c+\delta)$, 
$$f(c)-\epsilon<f(x)<f(c)+\epsilon.$$
Choose $0<\epsilon<f(c)$. What may we conclude?
