Is there significance in a function having an infinite number of derivatives?

For example, we can define a function

$$f(x) = \begin{cases} x^2 & \text{if } x < 2 \\ 4 \cdot(x - 1) & \text{if } x \ge 2 \end{cases}$$

This function seems to have a derivative.

$$f'(x) = \begin{cases} 2x & \text{if } x < 2 \\ 4 & \text{if } x \ge 2 \end{cases}$$

However, $$f$$ has no second derivative at $$2$$. Contrast this to $$\sin$$, which has an infinite number of derivatives. It seems that $$\sin$$ is somehow "smoother" than $$f$$.

Is there any significance to this?

• My understanding is that you're asking for some kind of intuition behind the difference between a $k$-differentiable map and an $m$-differentiable map. Could you make it more clear if this is not the case? Nov 29 '19 at 13:14
• What do you mean by $k$-differentiable maps? Nov 29 '19 at 13:15
• Diff $k$ times but not $k+1$. Nov 29 '19 at 13:16
• Yes, I was asking whether or not $k$-differentiability is important anywhere, and if $\infty$-differentiable maps are somehow a special class of functions. In fact, "$k$-differentiability" may actually be an answer, since having a name for it means it is at least somewhat significant Nov 29 '19 at 13:17
• Smooth functions are nice because they form the the largest class of functions on which differentiation is an operation into the same class. If you take a $k$ times differentiable function, its derivative needn't be $k$ times differentiable. But a derivative of an infinitely differentiable function is infinitely differentiable. Nov 29 '19 at 13:27

wojowu, Hans Lundmark and Arnaud Mortier answered the question in the comments: This is called smoothness, and $$\sin$$ is an infinitely smooth function.