Differentiability and Local Linearity

Differentiable functions have a property called local linearity, meaning that if is differentiable around and you zoom in far enough on , will appear linear. Use the applet below to observe this zooming for several different functions, including some with points where the function is not differentiable.

Were you able to find an example of a function that did not have the property of local linearity at a point? If so, which function (and which point)?

All differentiable functions have the property of local linearity, but so do some non-differentiable functions. Name one (and a corresponding point).