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Obviously intuitively the area under something that is above the x-axis is always positive, but how can I show this with a proof?

CIM
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    This is very likely a duplicate. – ncmathsadist Jan 29 '15 at 00:25
  • I've been searching and so far everything involved proving something is continuous based on the fact that it's Riemann integrable, which I don't need since I already know it's continuous. – CIM Jan 29 '15 at 00:26
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    Write down a lower Riemann sum. Show that such a sum is always positive. Hence the Riemann integral, which is greater than or equal to that lower sum must also be positive. – Simon S Jan 29 '15 at 00:29
  • Duplicates: https://math.stackexchange.com/q/553031/472818 – mr_e_man Dec 23 '22 at 03:08

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Choose a point $x$ at which $f(x) > 0$. Let $\epsilon = f(x)/2$. Choose $\delta > 0$ so that $|t - x| < \delta\implies f(x) > \epsilon.$ Can you do the rest?

ncmathsadist
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