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2 I have the following function $f(x)$ defined as $x^4sin(frac{1}{x})$ if $x neq 0$ and $0$ if $x=0$ . And I'm asked if the function is: a) differentiable b) two times differentiable c) two times continuously differentiable This function is of course quite similar to the function $x^2sin(frac{1}{x})$ , which can be found on the internet for the same type of exercises. But I still want to be sure that I proceeded in the correct way, because I have no solution to this exercise. Thanks for your feedback. a) At $x=0$ , we have $$lim_{hto 0}frac{f(0+h)-f(0)}{h}=lim_{h to 0}frac{h^4sin(frac{1}{h})}{h}=lim_{h to 0}h^3sin(frac{1}{h})=0$$ Everywhere else, we have simply $4x^3 sin(frac{1}{x})-x^2cos(frac{1}{x})$ . Thus the function is differentiable everywhere, even at $0$ where its value is $0$ . b) Again, ...