# Math Help - Continuous function of 2 variable

1. ## Continuous function of 2 variable

Assume $f(x,y)$ is continuous on $\{(x,y) | x>0, y\in\mathbb{R}\}$

for $\forall$ $y_{0}$

the limitation:

$\lim_{\substack{x\rightarrow 0^{+}\\y\rightarrow y_{0}}}f(x,y)=\varphi(y_{0})$

exists.

now we define function $g(x,y)$ as:

$g(x,y) = \begin{cases} f(x,y), & \mbox{if } x>0 \\ \varphi(y), & \mbox{if } x=0 \end{cases}$

show that:

$g(x,y)$ is continuous on $\{(x,y) | x\geq0, y\in\mathbb{R}\}$

2. Originally Posted by Xingyuan
Assume $f(x,y)$ is continuous on $\{(x,y) | x>0, y\in\mathbb{R}\}$

for $\forall$ $y_{0}$

the limitation:

$\lim_{\substack{x\rightarrow 0^{+}\\y\rightarrow y_{0}}}f(x,y)=\varphi(y_{0})$

exists.

now we define function $g(x,y)$ as:

$g(x,y) = \begin{cases} f(x,y), & \mbox{if } x>0 \\ \varphi(y), & \mbox{if } x=0 \end{cases}$

show that:

$g(x,y)$ is continuous on $\{(x,y) | x\geq0, y\in\mathbb{R}\}$
Where are you stuck?