# Thread: Continuous function of 2 variable

1. ## Continuous function of 2 variable

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

for $\displaystyle \forall$$\displaystyle y_{0} the limitation: \displaystyle \lim_{\substack{x\rightarrow 0^{+}\\y\rightarrow y_{0}}}f(x,y)=\varphi(y_{0}) exists. now we define function \displaystyle g(x,y) as: \displaystyle g(x,y) = \begin{cases} f(x,y), & \mbox{if } x>0 \\ \varphi(y), & \mbox{if } x=0 \end{cases} show that: \displaystyle g(x,y) is continuous on \displaystyle \{(x,y) | x\geq0, y\in\mathbb{R}\} 2. Originally Posted by Xingyuan Assume \displaystyle f(x,y) is continuous on \displaystyle \{(x,y) | x>0, y\in\mathbb{R}\} for \displaystyle \forall$$\displaystyle y_{0}$

the limitation:

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

exists.

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

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

show that:

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