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Let $f(x) = x|x|$. Then f(x) is: |
Continuous at x = 0 but not differentiable at x = 0 Differentiable at x = 0 but not Continuous at x = 0 Both Continuous and Differentiable at x = 0 Neither Continuous at x = 0 nor differentiable at x = 0 |
Both Continuous and Differentiable at x = 0 |
The correct answer is Option (3) - Both Continuous and Differentiable at x = 0 $f(x)=\left\{\begin{matrix}x^2,&x≥0\\-x^2,&x<0\end{matrix}\right.$ $f(0)=0,\lim\limits_{x→0^-}(-x^2)=0, \lim\limits_{x→0^+}(x^2)=0$ f(x) is continuous $f'(x)=\left\{\begin{matrix}2x,&x≥0\\-2x,&x<0\end{matrix}\right.$ $LHD=\lim\limits_{x→0^-}(-2x)=0=\lim\limits_{x→0^+}(2x)=RHD$ f(x) is differentiable |
