Smoothness

In mathematics, a smooth function is a function that can be differentiated indefinitely.

Smoothness Media

A bump function is a smooth function with compact support.
The C0 function $f$ ( $x$ ) = $x$ for $x$ ≥ 0 and 0 otherwise.
The function $g$ ( $x$ ) = $x$ 2 sin(1/ $x$ ) for $x$ > 0.
The function f:\R\to\R with f(x)=x^2\sin\left(\tfrac 1x\right) for x\neq 0 and f(0)=0 is differentiable. However, this function is not continuously differentiable.
A smooth function that is not analytic.
Two Bézier curve segments attached in a way that is only C0 continuous
Two Bézier curve segments attached in such a way that they are C1 continuous
Curves with G1-contact (circles,line)
(1-\varepsilon^2) x^2 -2px+y^2=0 , \ p>0 \ , \varepsilon\ge 0*pencil of conic sections with G2-contact: p fix, \varepsilon variable *(\varepsilon=0: circle,\varepsilon=0.8: ellipse, \varepsilon=1: parabola, \varepsilon=1.2: hyperbola)