average of derivative over function

Question

I have found this construction in calculus that i found useful. I wanted to find the average of the derivative over a real function from $-\infty$ to $+\infty$. I started with this intuitive integral: $$\lim_{h\rightarrow \infty}{1\over 2h}\int_{-h}^{h}f^{'}(x)dx$$ And i simplified it to this: $$=\lim_{h\rightarrow \infty}{f(h)-f(-h)\over 2h}=T[f(x)]$$ Is there any official name for this idea, or is it just another construction? It's simply the application of the formula for slope with limits approaching infinity. $$m={y_2-y_1\over x_2-x_1}$$

Here are some properties i have found with this construction. $$T[f(x)+g(x)]=T[f(x)]+T[g(x)]$$ $$f(-x)=f(x)\Rightarrow T[f(x)]=0$$ $$T[c_0x^0+c_1x^1+c_2x^2+c_3x^3+...]=T[c_1x^1+c_3x^3+c_5x^5+...]$$

Answer 1

For a non-decreasing function this is the same as 'total variation', and is the most similar functional I have come across.

Edit: your functional is not equal to the total variation in any case, but these seem to be related ideas