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Corollary: Sufficient Condition for a Function to be Constant
(related to Corollary: Estimating the Growth of a Function with its Derivative)
Let $a < b,$ $[a,b]$ be a closed real interval, and $f:[a,b]\to\mathbb R$ a continuous function, which is differentiable on the open real interval $]a,b[$ with $f'(x)=0$ for all $x\in]a,b[.$ Then $f$ is a constant function.
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- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983