Local linearity and differentiation

[apple2357]

Can someone help clarify my thinking on this please?

I understand that we can define the gradient of a curve ( say y=x^2) at a point on the curve using tangents because the curve has the property of local linearity. An intuitive ( not rigorous) way of seeing this is by zooming in using graphing software.

I recognise not every function has local linearity at a particular point , the only example i can think of is y= abs(x) at x=0, where there is a kink.

I also recognise there is a problem with curves like y= sqrt(9-x^2) at the point x=3, the curve looks linear but the gradient will be undefined as it is a vertical line. Does this still mean the curve is locally linear at x=3 even though we cant define its gradient?

And are there other examples of curves i can use to illustrate these special cases? And are there other strange examples i might have missed?

 

[JeffM]

First, look up Dirichlet’s function, which is defined for every real number, but is nowhere continuous. Obviously that function is not locally linear. For a less bizarre example, the signum function is not locally linear in any neighborhood that includes zero.

Second, I think in many cases what is “flat” is a matter of how we define things.

[MATH]z = x^2 + y^2[/MATH]
strikes me as ”flat” at every pair of real x and y values, including at those values in the neighborhood of z = 9, which certainly will include x = 3.

 

[apple2357]

Thanks JeffM.

Incidentally, I found a nice one:

y= x^3 + ((x−1)^(2/3))/7

Type this into graphing software and zoom in on (1,1)