Friday, February 3, 2023
HomeUncategorizedEvery monotonic function on 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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