I've long wondered how to "quantitatively" describe how smooth a function is.
For instance, a 1000 term Fourier series for a the Heaviside step function is technically smooth, as it has infinite derivatives. But, when compared to the first term, it is much "spikier".
I often want to find myself saying the 1000-term Fourier series is "less smooth" than the single term... but is there a good word for that?
One measure of "spikiness" (or maybe "wobblyness" would be more fitting) is the total variation. The formal definition is that $$ TV_a^b(f) = \sup_{a = x_1<x_2<\ldots x_n=b} \sum_{k=1}^n \left|f(x_{n+1}) - f(x_n)\right| \text{,} $$ meaning that you look at arbitrary fine partitions $[a,b] = [x_0,x_1] \cup [x_1,x_2] \cup \ldots \cup [x_{n-1},x_n]$ and sum up the amount by which $f$ varies between the start and the end of each subinterval.
If $f'$ exists and is riemann-integrable on $[a,b]$, then $$ TV_a^b(f) = \int_a^b |f'(x)| \,dx \text{,} $$ and it follows that for such $f$ you have that $$ TV_a^b(f) \leq (b-a)\sup_{x \in [a,b]} |f'(x)| \text{,} $$ meaning that the smaller a function's derivative is in $[a,b]$, the less "wobbly" that function is.
The dependency on the interval length is a bit unfortunate in that last inequality, but there's an easy remedy - just normalize with the interval length. So let's define the normalized wobblyness of $f$ on $[a,b]$ $$ W_a^b(f) = \frac{1}{b-a}TV_a^b(f) = \sup_{a = x_1<x_2<\ldots x_n=b} \sum_{k=1}^n \frac{\left|f(x_{n+1}) - f(x_n)\right|}{b-a} \text{,} $$ then $$ W_a^b(f) \leq \sup_{x \in [a,b]} |f'(x)| $$ if $f'$ exists and is riemann-integrable on $[a,b]$.
Related with the comment of Andrew D., you can be interested in read about the Runge phenomenon, in the section of constrained minimization. Maybe what you want is to limit the (absolute value of the) derivatives to be constrained in certain interval.
