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I know a differentiable function is strictly increasing on [a,b] if the derivative is positive on [a,b]. But what if the function is differentiable on the real line and the derivative is positive, can we then conclude that it is strictly increasing on $\mathbb{R}$? The mean value theorem is only stated for intervals, does it work on $(-\infty,\infty)$, or is there a counterexample to the statement?

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Yes, it does work on $(-\infty, \infty)$.

This is because if you take any $x,y\in\mathbb R$ such that $x<y$, you then know that $f$ is strictly increasing on $[x, y]$, and that tells you that $f(x)<f(y)$.

Since $x,y$ are arbitrary, you know that $f$ is increasing on $\mathbb R$.

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