Local linearity

Local linearity is a property of functions that says — roughly — that the more you zoom in on a point on the graph of the function (with equal scaling horizontally and vertically), the more the graph will look like a straight line. More precisely, a function is locally linear at a point if and only if a tangent line exists at that point. The idea of local linearity is often introduced as a picture of what it means for a function to be differentiable at a point.

Functions are locally linear everywhere except

• Where they have a discontinuity. That is, jumps, breaks, vertical asymptotes, etc.
• Places where the function has "sharp corners", or cusps. That is the function f(x) is continuous at x = a but

the one-sided derivatives $\lim_{h \to 0^+} \frac{f(a+h)-f(a)}{h}$ and $\lim_{h \to 0^-} \frac{f(a+h)-f(a)}{h}$ are unequal or undefined.

Functions that are locally linear have graphs that appear smooth; but they need not be smooth in the mathematical sense, which requires that the function be differentiable infinitely many times. A function that is only once differentiable at a point is locally linear there. However, a function with a vertical tangent line will be locally linear but not differentiable, because the slope of the tangent line is undefined. For example $y=\sqrt[3]{x}$ is locally linear at the origin but is not differentiable there. Thus, a function that is locally linear at a point will be differentiable there unless it has a vertical tangent line at said point.