f(x) = x^2 sin (1/x) if x in (-1,1) and x is not 0
f(x)=0 if x=0
Investigate the validity of Taylor's theorem for f about the point x=0
An $\displaystyle f(x)$ can be expanded in Taylor series around $\displaystyle x=x_{0}$ if in that point $\displaystyle f(x)$ and all its derivatives do exist. For the function You have proposed, in $\displaystyle x=0$, $\displaystyle f(x)$ does exist but $\displaystyle f^{'}(x)$ doesn't ...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$