TITLE: smooth functions or continuous
QUESTION [20 upvotes]: When wesay a function is smooth?
Is there any difference between smooth function and continuous function?
If they are the same, why sometimes we say f is smooth and sometimes f is continuous?

REPLY [2 votes]: Take $f(x) = x|x|$ it is smooth, and now consider $g(x) = x^3$, this other function is also smooth. However, $g(x)$ is much smoother than $f(x)$ because derivitive of $f(x)$. You can argue that $g(x)$ is infinitely many times smooth. All polynomials belong to $C^\infty$ meaning they are infinitely many times differentiable and are smooth.
However, $h(x) = |x|$ is not smooth, because it has corner.
Please note that all three functions, $f(x)$, $g(x)$, and $h(x)$ are continous. 
Here is how $f(x)$ looks like:

Here is how the derivative of $f(x)$ looks like:

Here is the second derivate of $f(x)$, as you can see its second derivative is not even continuous: 

Here is the graph of $g(x)$:

Here is graph of $\frac{d f(x)}{dx} = 3x^2$: 

Here is graph of $\frac{d^2 g(x)}{dx^2} = 6x$: