# how to prove continuity in multivariable functions?

$$\frac{x^3y}{x^4+y^2},etc.,$$ in these multi-variable functions its easy to prove discontinuity by giving counterexamples but for proving continuity are there any tricks?

using $\epsilon$ definitions seems to so tougher in multivariable functions compared to single variable calculus

• say at which point u can use epsilon definition also – user106301 Nov 8 '13 at 1:40

You're interested in continuity at the point of $$\left ( 0,0 \right )$$.
Use the squeeze theorem $$\left| \frac{x^3y}{x^4+y^2} \right| \le \left| \frac{x^3y}{2x^2y} \right| = \frac{1}{2}\left|x \right| \to 0$$ for $$\left ( x,y \right) \to \left ( 0,0 \right)$$