# Can differentiability conditions prevent arbitrarily small “ripples”?

Functions like $f_p(x)=x^p\sin(1/x)$ with $p\in\mathbb{Z}^+$ have removable discontinuities at $0$ and interesting behavior in any neighborhood of $0$. In particular, we see smaller and smaller "ripples" as we approach $0$, and these ripples never disappear, regardless of how far we zoom into the picture.

While I realize that the notion of local linearity is not a formal concept, it seems like we're violating that idea here? But as far as I can tell, sufficiently large values of $p$ allow $f_p$ to be as differentiable as we would like... Am I missing something?

The main question: Can I impose sufficient disability conditions to ensure that I actually do see "local linearity"?

I'm quite unsure about what terminology I should be using here, so please correct me if there's a better way say what I'm describing.

(It seems like this might be related to bounded variation, but it's unclear to me how.)

• What do you mean by local linearity? The derivative is locally the best linear approximation – user251257 Aug 27 '16 at 21:57
• I'm talking about the super-soft "high school calculus" idea; that is, "the function looks like a line when you zoom in far enough." – AegisCruiser Aug 27 '16 at 22:03
• that is nonsense!? – user251257 Aug 27 '16 at 22:12
• @user251257 No it's not nonsense. – zhw. Aug 27 '16 at 22:37