Proving Continuity and Inequalities for a Limit with Two Variables

[oahsen]

Homework Statement

Let
f(x, y) =((x^2)*y)/(x^4 + y^2) if (x, y) != (0, 0) ,
f(x,y) = 0 if (x, y) = (0, 0) .
a) Is f continuous at (0, 0)? Prove your statement.
b) Show that
-1/2 ≤ f(x, y) ≤1/2
for all (x, y).

I have used the two path test to show that it has not limit at 0,0 hence it is not continuous there. however, ı have no idea what I should do for the b part?Is there a algebratic way to show it or should we take differential etc.?

 

[daveb]

Try solving for each inequality separately, and cross multiply to get a perfect square.

 

[oahsen]

Try solving for each inequality separately.

Are you referring to the fx and fy (the partial derivatives of f) by saying "for each inequality"?

 

[Data]

What he means is to find necessary and sufficient conditions on x,y for each of the two inequalities

-1 \leq \frac{2x^2y}{x^4+y^2}

and

1 \geq \frac{2x^2y}{x^4+y^2}

(ie. rearrange them, using "reversible" steps, until you find something that will tell you for which x,y they are satisfied)