Question

How do I prove that the xsin(1/x) is continuous at x doesn't equal 0?

Answer 1

must be a definition of continuity you can use

Answer 2

if the limits at f(a) is equal to f(a), then it is continuous at x=a, is what im recalling as a definition. not sure if there are others that would be more suitable

Answer 3

\[sin(u) =\sum_0\frac{(-1)^n}{(2n-1)!}u^{2n-1} \]
\[sin(1/x) =\sum_0\frac{(-1)^n}{(2n-1)!}x^{1-2n} \]
\[x~sin(1/x) =\sum_0\frac{(-1)^n}{(2n-1)!}x^{-2n} \]