Continuity of multivariable piecewise function (cos, sin)

Question

Continuity of multivariable piecewise function (sin, cos)

Let $$ f(x,y) = \begin{cases} \dfrac{\cos(x)xy⁴ + a\sin(x⁴)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\ 0\quad& \text{if} \quad (x,y)=(0,0). \end{cases} $$

I should analyze continuity of $f$ in $(0,0)$ for $a = 0$ and $a = 1$. Exercise says that I must justify my answer.

First step) For $a = 0$,

$$ f(x,y) = \begin{cases} \dfrac{\cos(x)xy⁴}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\ 0\quad& \text{if} \quad (x,y)=(0,0). \end{cases} $$

Step 2: For $a = 1$ $$ f(x,y) = \begin{cases} \dfrac{\cos(x)xy⁴ + a\sin(x⁴)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\ 0\quad& \text{if} \quad (x,y)=(0,0). \end{cases} $$

Step 3: I try to solve for $a = 0$. $f(x,y)$ is continuous at $(0,0)$ $ \iff $ $\lim_{x,y \rightarrow 0,0} f(x)= f(0,0) $ In this case, $f(0,0) = 0$.

Step 4 (for $a = 0$):

$\lim_{x,y \rightarrow 0,0} f(x)= \lim_{x,y \rightarrow 0,0} \dfrac{cos(x)xy⁴}{(x^2 + y^2)}\quad $

Limit definition from Wikipedia says

for every ε > 0 there exists a δ > 0 such that for all (x,y) with 0 < ||(x,y) − (p,q)|| < δ, then |f(x,y) − L| < ε

In this case, $L = 0$, and $p=0=q$.

The problem is that I do not know how to go on. cos(x) makes me confused. What should I do? I really think we should need something like this but for two variables. Thanks in advance.

Answer 1

No need to go back to the definition of limits, use some of the properties instead. If $f(x,y)$ is continuous at $(0,0)$, consider $p(x) = (x, 0)$; $p$ is continuous everywhere (perhaps you need to prove that, but that's easy). So: if $f$ is continuous at $(0,0)$, then $g(x) = f \circ p (x)$ is continuous at $0$. Compute $f \circ p$ and show it is not continuous at $0$ for either $a=0$ or $a=1$, again, this is much easier than the original problem. So the conclusion is that the original $f$ is not continuous at $(0,0)$ for either value of $a$.

I gave the argument in relatively formal terms; but what I am doing is this: If $f$ is continuous as a function of two variables, it should in particular be continuous when I fix $y=0$ - and it just isn't.