Definef (x) =(x2 if x is rational,0 if x is irrational.Show that f is differentiable at 0 and f ′(0) = 0
Question
Solution 1
To show that the function f is differentiable at 0, we need to show that the limit as x approaches 0 of [f(x) - f(0)] / x exists.
First, let's note that f(0) = 0^2 = 0.
If x is rational, f(x) = x^2. So, [f(x) - f(0)] / x = [x^2 - 0] / x = x.
If x is irrational, f(x) = 0. So, [f(x) - f(0)] / x = [ Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
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