# Understand the problem

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

##### Source of the problem

TIFR GS 2017 Entrance Examination Paper

##### Topic

Real Analysis

##### Difficulty Level

Easy

##### Suggested Book

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.

# Start with hints

Do you really need a hint? Try it first!

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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