exercise:8f84d58d25

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Consider a function [math]f[/math] which is integrable over [math][a,b][/math] and which, in addition, satisfies:

1. [math]f[/math] is continuous at every point of [math][a,b][/math].
2. [math]f(x) \geq 0[/math], for every [math]x[/math] in [math][a,b][/math].
3. [math]f(c) \gt 0[/math] for at least one point [math]c[/math] in [math][a,b][/math].

Prove that [math]\int_a^b f(x) \; dx \gt 0[/math].