Differentiable functions have a property called [i]local linearity[/i], meaning that if [math]f[/math] is differentiable around [math]x=a[/math] and you zoom in far enough on [math](a,f(a))[/math], [math]f[/math] will appear linear.[br][br]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)?
Yes. For example, [math]y=\left|x\right|[/math] at [math]x=0[/math]. Also, a less popular choice: [math]y=\frac{\left|x\right|}{x}[/math] at [math]x=0[/math].
All differentiable functions have the property of local linearity, but so do some non-differentiable functions. Name one (and a corresponding point).
[math]f(x)=x^{\frac{1}{3}}[/math] is not differentiable at [math]x=0[/math], and yet it appears locally linear at [math]x=0[/math].