So I have a function of two variables f(x,y)=y^x. where is this function continuous? I'm thinking that the only restriction that applies is y>0

Answer 1

if y > 0, then y^x is defined for any real value of x
e.g. 4^12, 5.4^(1/7), 3^0 .....

if y < 0 (the base is negative), the exponent cannot be a fraction with an even base, (taking an even root of a negatiave number)
e.g. (-4)^(-1/2) is not defined, but
(-4)^(2/3) is defined.

Also 0^0 is not defined.

explanation:
5^0 = 1
7^0 = 1
so following that pattern 0^0 should be 1
and
0^2 = 0
0^5 = 0
so following this pattern 0^0 should be zero
AARGHHH! A mathematician's nightmare, the same calculation producing 2 different answers.
Better call that one undefined!

I think you should have enough information here.