Continuity of multivariable functions

mreaume
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Homework Statement



A function f is defined on the whole of the xy-plane as follows:

f(x,y) = 0 if x=0
f(x,y) = 0 if y = 0
f(x,y) = g(x,y)/(x^2 + y^2) otherwise

a) g(x,y) = 5x^3sin(y)
b) g(x,y) = 6x^3 + y^3
c) g(x,y) = 8xy

For each of the following functions g determine if the corresponding function f is continuous on the whole plane

Homework Equations



A function's limit exists if and only if it is not dependent of the path taken.

The Attempt at a Solution



Since the functions are continuous for all values of x and y, the only restriction on the xy plane is at the point (0,0). So I am trying to find the limit of these functions as (x,y) approaches (0,0).

I have done so for c) using the line x=0 and y=x. These produce two different answers. Therefore, the limit of c does not exist at (0,0) and the function is not continuous on the xy plane.

Any tips as to how I should tackle the other ones? I suspect that their limits are 0 (since every path I try gives 0), but I am having a hard time proving this.
 
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Aren't they continuous at any point where the partial derivatives exist?
 
Thanks. Turns out I misread the question. I ended up using your tip (partial derivatives) and was able to solve the problem.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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