Local linearity
Encyclopedia
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
the one-sided derivatives
and 
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
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.
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
. 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 cuspCusp (singularity)In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
s. That is the function
is continuous at 
but
the one-sided derivatives
and are unequal or undefined.
Functions that are locally linear have graphs that appear smooth; but they need not be smooth
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
in the mathematical sense, which requires that the function be differentiable infinitely many times. A function that is only once differentiable
Differentiable function
In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...
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
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.