True or False? If \( f: \Bbb R \to \Bbb R \) is differentiable and bijective then \(f^{-1} \) is also differentiable.

TIFR GS 2017 Entrance Examination Paper

Real Analysis

Easy

Introduction to Real Analysis, Fourth Edition, English, Paperback, by Robert G. Bartle and Donald R. Sherbert. Introduction to Real Analysis, Seventh Edition, English, Paperback, by S. K. Mapa.

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The given statement is a general statement. So, if I can find a function \(f(x) \) which is differentiable and bijective on \( \Bbb R\) but whose inverse function \( f^{-1}\) is not differentiable then I would be able to show that the statement is not generally true. Now, can you find such a function?

Let us consider the function \(f(x) = x^3 \) . Can you now show that the function is contradicting the statement given to us?

The function \( f(x)= x^3 \) is a bijective and differentiable function. It’s inverse is \( x^{\frac{1}{3}} \) which, you can see, is not differentiale at \( x=0\). Hence the statement is false.

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