TITLE: If $f:[a, b] \to \mathbb{R}$ is a continuous function, s.t. $f(x)>0$ for all $x$, prove that there exists $c >0$, s.t. $f(x) > c$ for all $x$.
QUESTION [5 upvotes]: If $f:[a, b] \to \mathbb{R}$ is a continuous function, s.t. $f(x)>0$ for all $x$, prove that there exists $c >0$, s.t. $f(x) > c$ for all x.
I am not sure how to approach this problem, I have tried proving this by using IVT but no results so far.

REPLY [4 votes]: $f$ continuous on $[a,b]$, $f$ attains its minimum , i.e there is a $m \in [a,b]$   such that $f(x)  \ge f(m) \gt 0.$
For $c:= (1/2)f(m)$ we have:
$f(x) \gt c,$ for $x\in [a,b].$