Example of discontinuous function with partial derivatives
Define the function f of two variables by
f (x, y) =  |
0 |
if (x, y) = (0, 0) |
| |
if
(x, y) ¹ (0, 0). |
This function has partial derivatives with respect to x and with respect to y for all values of (x, y). (For every fixed value of y the function gy defined by gy(x) = f (x, y) for all x is differentiable, and
for every fixed value of x the function hx defined by hx(y) = f (x, y) for all y is differentiable. Note that for g0(x) = 0 for all x and
h0(y) = 0 for all y.) However, f is not continuous at (0, 0): we have f (0, 0) = 0, but f (x, x) = 1/2, for example, for all x ¹ 0.
Return to main page at point example is mentioned
Copyright © 2003 by Martin J. Osborne