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

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.

Answer 1

$f$ 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 $x\in [a,b].$