Math 13 Fall 2004

Calculus of Vector-valued Functions

Example of a function that has

both partial derivatives at (0,0), but is not differentiable

October 4, 2004

   Define a scalar-valued function of two variables

>	f := (x, y) -> x^3 / (x^2 + y^2);

    Have a look at its graph

>	plot3d(f(x, y), x = -1..1, y = -1..1);

    f seems to be continuous everywhere and indeed it is.

    But f is not differentiable at (0, 0) (see the handout for details).

      Let's plot its partial derivatives

>	plot3d(diff(f(x, y), x), x = -1..1, y = -1..1); plot3d(diff(f(x, y), y), x = -1..1, y = -1..1);

     They are obviously discontinuous!!!

>	factor(diff(f(x, y), x)); factor(diff(f(x, y), y));

>