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 $\!(a,b)$, $(a,\infty )$, $(-\infty ,b)$, 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.:

$x_{1},x_{2}\in I,x_{1}<x_{2}\implies f(x_{1})<f(x_{2})$

Related facts

Similar facts

• Zero derivative implies locally constant
• Negative derivative implies decreasing

Converse

• Increasing and differentiable implies nonnegative derivative that is zero only at isolated points

Facts used

1. Lagrange mean value theorem

Proof

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