Local linearity

Local linearity

Local linearity is a property of differentiable 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 with a slope equal to the derivative of the function at the point.

Thus, local linearity is the graphical manifestation of differentiability.

Functions that are locally linear are smooth. Functions are not locally linear at points where they have discontinuities: breaks, jumps, vertical asymptotes, or the like. For example, y = |x|, is not locally linear at the origin.

Functions that are differentiable at a point are locally linear there, and vice versa. Unfortunately, there is no other definition of local linearity. It would be helpful if one could determine whether a function is locally linear at a point and then be able to conclude that is it differentiable. This is not possible. Since "zooming in" is accomplished using technology, such as a graphing calculator or computer graphing program, one can never be certain that one has zoomed in far enough. Thus if your function is given to you numerically, you can never be sure.

On the other hand, if you have an analytic representation of your function (i.e. a formula containing standard functions like powers, trigonometric functions, etc.), then you can check to see if the function is differentiable at the point of interest. Namely, take the derivative at the point of interest and see if it is finite, and if so, it is differentiable and hence locally linear.

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