4

Let $f(x,y)=\frac{x^2y}{x^4+y^2} $ if $(x,y) \ne (0,0)$ and $f(0,0)=0$ if $(x,y)=(0,0)$

This is a question from a university entrance exam. Generally i get stuck in these types of problems where the numerator and denominator powers are tough to cancel out and using the polar coordinates makes it more complicated.(Atleast on my part)

I came across the solution which went like this:

$m=2,n=1,i=4,j=2 (even)$ and $mj+ni=ij$ so the function is not continuous at $(0,0)$

Does anyone have any idea behind the logic of this trick?

ThirstForMaths
  • 1,972

1 Answers1

3

Note that $$\lim_{y \to 0} f(y,y^2)= \lim_{y \to 0} \frac{y^4}{2y^4}=\frac{1}{2}$$ but $f(0,0)=0$ . Hence $f$ is not continuous at $0$.

I mentioned the general exercise you want ( It somewhat helpful ), but I dont know the solution

Exercise: [Source: A course in Multivariable calculus and analysis by Ghorpade, Page 79, problem 15]

Let $m, n$ be nonnegative integers and let $i, j \in \Bbb N$ be even. Let $f : \Bbb R^2 \to \Bbb R$ be defined by $f(0, 0) := 0$ and $$f(x, y) := \frac{x^my^n}{x^i + y^j}$$ for $(x, y) \neq (0, 0)$. Show that $f$ is continuous at $(0, 0)$ if and only if $mj + ni > ij$

Chinnapparaj R
  • 11,589
  • 1
    [+1] for aligning your answer to the OP's question. – A. P. Jan 16 '23 at 10:39
  • 2
    Sandwich should be work (sketch): $|f(x,y)|\leqslant (x^{i}+y^{j})^{(mj+nj-ij)/ij}\implies f(x,y)\to (0,0) \iff mj+ni-ij>0$ for $i, j,m,n>0$. – A. P. Jan 16 '23 at 11:04