# Positive derivative implies increasing

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## Statement

### On an open interval

Suppose $f$ is a function on an open interval $I$ that may be infinite in one or both directions (i..e, $I$ is of the form $(a,b)$, $(a,\infty), [itex](-\infty,b)$, or $(-\infty,\infty)$). Suppose the derivative of $f$ exists and is positive everywhere on $I$, i.e., $f'(x) > 0$ for all $x \in I$. Then, $f$ is an increasing function on $I$, i.e.:

$x_1, x_2 \in I, \qquad x_1 < x_2 \implies f(x_1) < f(x_2)$

## Facts used

1. Lagrange mean value theorem

## Proof

Fill this in later