Question 7.T.11: (Inverse Function Theorem) Let f be a differentiable functio...

Elements of Real Analysis

(Inverse Function Theorem)

Let f be a differentiable function whose derivative is continuous on the open interval I. If $f^{′}(b) ≠ 0$ for some b ∈ I, then there is an open interval J ⊆ I which contains b such that $f |_{J}$ is injective, and the function $( f |_{J} )^{−1}$ is differentiable on f(J), where its derivative is

$((f |_{J} )^{−1})^{′} (f(x)) = \frac{1}{f^{′}(x)}$ for all x ∈ J.

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