Special Functions Differentiability Chart

Function	At x=0	Continuous?	Differentiable?	f'(0)
\|x\|	Corner	Yes	No	DNE
x\|x\|	Smooth	Yes	Yes	0
$x^2$ \|x\|	Smooth	Yes	Yes (twice)	0
x sin $\frac{1}{x}$	Oscillating	Yes	No	DNE
$x^2$ sin $\frac{1}{x}$	Oscillating	Yes	Yes	0
$x^3$ sin $\frac{1}{x}$	Oscillating	Yes	Yes	0
[x]	Jump	No	No	DNE
{x}	Jump	No	No	DNE
sgn(x)	Jump	No	No	DNE
x^ $\frac{1}{3}$	Vertical tangent	Yes	No	infinity
x^ $\frac{2}{3}$	Cusp	Yes	No	+/-inf
e^(-1/ $x^2$ ), f(0)=0	Flat	Yes	Yes	0