Prove that $f\colon \Bbb R^2\to \Bbb R$ is continuous

Asked Oct 19 '13 at 22:58

Active Oct 20 '13 at 00:47

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Let $f\colon \mathbb{R}^2 \to \mathbb{R}$ be function, such that for every fixed $x_0$

$f(x_0,y): \mathbb{R} \to \mathbb{R}$ is monotone and continuous function

and for every fixed $y_0$

$f(x,y_0): \mathbb{R} \to \mathbb{R}$ is continuous function.

Prove that $f$ is continuous.

I am trying to check the continuity at $(x',y')$, So let $\epsilon > 0$ and now $|f(x,y)-f(x',y')| \le |f(x,y)-f(x,y')|+|f(x,y')-f(x',y')|$ by triangle inequality (I'm trying to move colloterally to axises). Now I think I should use the continuity of $f(x_0,y)$ and $f(x_0,y)$ in apriopriate way, but I don't se how.

edited Oct 19 '13 at 23:52

asked Oct 19 '13 at 22:58

amoneth

1

This was asked recently. – Git Gud Oct 19 '13 at 23:02
Thanks. Do you remember how it was tagged? I can't find it in real-analysis :( – amoneth Oct 19 '13 at 23:20
I don't remember anything specific that could help you find the question. Sorry. You can try the tag multivariable-calculus. – Git Gud Oct 19 '13 at 23:22
http://math.stackexchange.com/questions/524631/continuous-function – njguliyev Oct 20 '13 at 07:48

Prove that $f\colon \Bbb R^2\to \Bbb R$ is continuous

0 Answers0