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 $(a,b)$, $(a,\mathrm{\infty })$, $(-\mathrm{\infty },b)$, or $(-\mathrm{\infty },\mathrm{\infty })$). Suppose the derivative of $f$ exists and is positive everywhere on $I$, i.e., $f^{\prime }(x)>0$ for all $x\in I$. Then, $f$ is an increasing function on $I$, i.e.:

$\mathrm{\forall }{x}_1,x_2\in I,x_1<x_2⟹f(x_1)<f(x_2)$

Facts used

1. Lagrange mean value theorem

Proof

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