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A follow up question of the cited question bellow.

Consider a function $f(x, y):\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is continuous in each argument. If we have continuous function $y(x):\mathbb{R}\rightarrow\mathbb{R}$, is $g(x)\equiv f(x,y(x))$ continuous?

In the former question, the joint continuity fails with a counter example. In the example the basic idea is to construct a path along which the function is not continuous, yet if we set $y(x)$ fixed, there is only one path to be verified. So is there maybe some conditions under which $f(x,y(x)$ is continuous?

@MISC {98831, TITLE = {Continuity and Joint Continuity}, AUTHOR = {user12586 (https://math.stackexchange.com/users/12586/user12586)}, HOWPUBLISHED = {Mathematics Stack Exchange}, NOTE = {URL:Continuity and Joint Continuity (version: 2012-01-13)}, EPRINT = {Continuity and Joint Continuity}, URL = {Continuity and Joint Continuity} }

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  • Did you try to look at the example given on the other page? – Did Sep 24 '15 at 11:36
  • Yes, I did. Basically I'm rather curious if there's a general rule with which we can find the path that the continuity fails, or that we can only learn every case by constructing a counter example by experience. – MOONkey Sep 24 '15 at 11:42
  • Then the question should probably be closed as "too vague", rather than as a duplicate. But unless you specify the kind of condition you have in mind... – Did Sep 24 '15 at 11:46
  • Yes, you are right. New user here, thanks for the comment. I'll pay more effort on it myself then ask again. – MOONkey Sep 24 '15 at 11:51

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