Riemann Integrable

1 min read Updated Fri Apr 24 2026 03:19:45 GMT+0000 (Coordinated Universal Time)

A bounded function $f:[a,b]\rightarrow\mathbb{R}$ is Riemann integrable on $[a,b]$ iff $U(f)=L(f)$ . In that case, the Riemann integral of $f$ on $[a,b]$ is denoted by:

\int_{a}^{b}{f(x)\,\text{d}x}

Reimann Integrable or not

Function	Yes or No?	How?
Unbounded	No	By definition
Constant	Yes	$\forall P\,\text{(any partition)}\;\; L(f;P)=U(f;P)$
Monotonically increasing/decreasing	Yes	Take a partition such that $\Delta{x}<\delta=\frac{\epsilon}{f(b)-f(a)}$
Continuous	Yes	Take a partition such that $\Delta{x}<\delta=\frac{\epsilon}{2(b-a)}$

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