# Theorem :

## Every monotonic function on [a, b] is Riemann integrable.

# Proof :

Let $f:[a, b]\to \mathbb{R}$ be monotonic function.

If f is constant function then obviously it is Riemann integrable.

WLOG, suppose f is strictly increasing.

$\therefore f(a)<f(x_1)<f(x_2)<f(b)$, $a<x_1<x_2<b$

$\therefore f(b)-f(a)>0$

Choose a partition of [a, b] for $\epsilon>0$ such that,

$P=\{a=x_0, x_1, x_2, ... , x_n=b\}$ with $\|P\|<\frac{\epsilon}{f(b)-f(a)}$

for $[x_{k-1}, x_k], \|x_k-x_{k-1}\|<\frac{\epsilon}{f(b)-f(a)}$

Consider,

$U(f, P)-L(f, P)$

$=\displaystyle\sum_{k=1}^{n}(M_k-m_k)\Delta_{x_k}$

$=\displaystyle\sum_{k=1}^{n}[f(x_k)-f(x_{k-1})]\Delta_{x_k}$

$\leq\displaystyle\sum_{k=1}^{n}[f(x_k)-f(x_{k-1})]\|P\|$

$<\displaystyle\sum_{k=1}^{n}[f(x_k)-f(x_{k-1})]\frac{\epsilon}{f(b)-f(a)}$

$=\frac{\epsilon}{f(b)-f(a)}\displaystyle\sum_{k=1}^{n}[f(x_k)-f(x_{k-1})]$

$=\frac{\epsilon}{f(b)-f(a)}\times[f(b)-f(a)]$

$=\epsilon$

$\therefore U(f, P)-L(f, P)<\epsilon$

$\therefore$ by Riemann criterion,

f is Riemann integrable.

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