Exercise · ZW-INTEGRABILITY-OF-CONTINUOUS-FUNCTIONS-010 · ★★★ Advanced
Let \(f\ge0\) be continuous on \([0,1]\) with \(\displaystyle\int_0^1 f=0\). What follows?
Choose one option.
Show answer
- A. \(f(x)=0\) for all \(x\in[0,1]\)
Explanation
If \(f(c)>0\), continuity gives \(f>f(c)/2\) on an interval around \(c\), and the integral would be positive. Hence \(f\) is identically zero.
Review the concept: Integrability of continuous functions
Result stating that every function \(f\) continuous on a closed, bounded interval \([a,b]\) is Riemann integrable there.
\[f\text{ continuous on }[a,b]\ \Rightarrow\ \int_a^b f(x)\,dx\text{ exists}\]
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