Positive derivative implies increasing

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 $(a,b)$, $(a,\infty ),<math>(-\infty ,b)$, 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

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