Local linearity

Local linearity is a property of functions that says — roughly — that if you zoom in on a point on the graph of the function (with equal scaling horizontally and vertically), the graph will eventually look like a straight line. More precisely, a function is locally linear at a point if and only if a tangent line exists at said point.

Thus, local linearity is the graphical manifestation of differentiability.

Functions that are locally linear are called smooth. 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". That is the function $f(x)$ is continuous at $x=a$ but

the right hand limit $\lim _{h\to 0^{+}}{\frac {f(a+h)-f(a)}{h}}$ and the left-hand limit $\lim _{h\to 0^{-}}{\frac {f(a+h)-f(a)}{h}}$ are unequal.

Functions that are differentiable at a point are 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.