The continuity of f(x,y)=ln(x^2+y^2)

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Question from anu, a student:

the question says we have to find the points in the plane where the function is continuous:
f(x,y)=ln(x^2+y^2) . here we aren't given a particular point (x,y) where we have to check a function's
continuity.
what is to be done if we have to check continuity over the whole domain of the function?
please help .

Anu,

$f(x,y)$ is the composition of two functions. If $g(x,y) = x^2 + y^2$ and $h(s) = \ln(s)$ then $f(x,y) = h(g(x,y)).$ $g(x,y)$ is continuous for all $x$ and $y.$ What is the range of $g(x,y)?$ For what values of $s$ in the range of $g(x,y)$ is $h(s)$ not continuous?

Harley

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