Wednesday, 13 April 2016

calculus - Paradox: Is the derivative of this function continuous at x=0?

h(x)={x2sin(1x) if x00 if x=0


Is the derivative of h(x) continuous at x=0?



How about the derivative of k(x)=xh(x)?



When I tried to differentiate it directly, the cos(1/x) in the result suggests that the limit of h(x) when x0 does not exist.

However, when I use the definition to calculate the derivative, it shows that the derivative is 0 when x0.




Which one is correct?

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