$\displaystyle suppose \ that \ f:[a.b] \to R \ is \ continuous \ with \ f(x)>0 \ , \forall x \in \ [a,b]

$ $\displaystyle . prove \ that \ \exists c>0 \ such \ that \ f(x) >c , \forall x \in [a,b] $

Do you know that a continuous attains its minimum and its maximum of a closed interval?
If so then $\displaystyle \left( {\exists d \in [a,b]} \right)\left( {\forall x \in [a,b]} \right)\left[ {f(x) \geqslant f(d) >\frac{{f(d)}}{2} > 0} \right]$