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Let f : [a,b] → R be a continuous function such that f(x) doesn't equal 0...

Let f : [a,b] → R be a continuous function such that f(x) doesn't equal 0 for every x ∈ [a,b].

1) Show that either f(x) > 0 for every x ∈ [a,b] or f(x) < 0 for every x ∈ [a,b].

2) Assume that f(x) > 0 for every x ∈ [a,b] and prove that there exists ε > 0 such that f(x) ≥ ε for all x ∈ [a,b].

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