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Show that function is not continuous by showing the limit does not exist.

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Recall that if a function f of one variable is differentiable at x_0, it has a derivative at x_0 (via the limit definition of derivative). As one consequence, we know that f is continuous at x_0. This is not necessarily so in the case of a function of two or more variables. Consider the function:

f(x,y) = (xy^2) / (x^2 + y^4) where (x,y) =/ (0,0)

f(x,y) = 0 where (x,y) = (0,0)

d. Show that f(x,y) is not continuous at (0,0) by showing that lim f(x,y) as (x,y) --> (0,0) does not exist.

Thus, we have shown that f(x,y) has partial derivatives for all (x,y) but is not continuous at (0,0).

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  • MSc, Meerut University
  • MPhil, Institute of Advanced Studies
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