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$$\frac{x^3y}{x^4+y^2},etc.,$$ 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

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

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