Question

find out if the following function is continuous:

\[\frac{x}{\sqrt{y}}\] if y > 0
0 if y = 0

also, find out if it is uniformly continuos in D = { (x,y) : |x| <= y <= 2 }

What I've done so far:

the function to be continuos the limit to (0,0) must match the f(0,0)

Anyhow, I found out the limit is path dependant, because if I run the limit through the line

y = x

I have the limit to be 0

But if I run it through:

y = x^2

The limit is 1.

I'm not sure if this is correct or not. Anyhow, how do I determine its uniform continuity ?