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**acevipa** I'm having some trouble with this question:

$\displaystyle Let f : [a,b] \rightarrow \mathbb{R} $ $\displaystyle be\ a\ continuous\ function\ such\ that\ f(x)>0$ $\displaystyle for\ all\ x \in [a,b].$ $\displaystyle Show\ that\ there\ is\ a\ positive$ $\displaystyle constant\ c\ such\ that\ f(x) \geq c$ $\displaystyle for\ all\ x \in [a,b].$ $\displaystyle Give\ examples\ to\ show\ that\ the\ conclusion\ fails$ $\displaystyle if\ [a,b]\ is\ replaced\ by\ (0,1] or [0,\inf ).$