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Let $u=f(z)$ be an elementary function and $z=g(x,y)$ a two-variable elementary function. Suppose we have $g(x,y)$ is continuous at point $(a,b)$, $g(a,b)=c$, and $\lim_{z\to c}f(z)=A$. When is it true that $\lim_{(x,y)\to(a,b)}f(g(x,y))=A$? If it's not true in general, are there any counter examples?

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1 Answer 1

The fact that $g$ is a function of two variables is totally irrelevant. Anyway, you are asking for the composition law for limits. Of course, the answer is positive is $f$ is continuous at $c$, since the composition of two continuous maps is always a continuous map. If $\lim_{z \to c} f(z)$ merely exists but $f$ is not necessarily continuous, the answer is negative, in general. The trouble comes from the fact that $g$ may be constant and equal to $c$. In this case, $f \circ g$ is constant and equal to $f(c)$ in a neighborhood of $(a,b)$, but $A$ may differ from $f(c)$. I am assuming that $f(c)$ is defined, of course.

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In the question, I assume both $f$ and $g$ to be elmentary functions, therefore if $f$ is defined at $c$, it's continuous at $c$ and the question is always true. –  Z.H. Apr 26 '12 at 9:40
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Sorry, but I do not agree. A function with a jump discontinuity is, in my opinion, an elementary function (you paste for example two constant functions), but it is discontinuous. If you already know that your functions are continuous, you should also know that compositions preserve continuity, shouldn't you? –  Siminore Apr 26 '12 at 12:18

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