- #1
Gooolati
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Homework Statement
Prove:
f(x,y) = [itex]\frac{x(x^{2}-y^{2}}{(x^{2}+y^{2}}[/itex] if (x,y) [itex]\neq[/itex] (0,0)
0 if (x,y) = (0,0)
is continuous at the origin
Homework Equations
[itex]\forall[/itex] [itex]\epsilon[/itex] > 0 [itex]\exists[/itex] [itex]\delta[/itex] > 0 s.t. if |(x,y)| < [itex]\delta[/itex] then |f(x,y)| < [itex]\epsilon[/itex]
(Since we are proving continuity at the origin)
The Attempt at a Solution
|(x,y)| < [itex]\delta[/itex] [itex]\Leftrightarrow[/itex] x[itex]^{2}[/itex] + y[itex]^{2} < \delta^{2}[/itex]
then this means that |x[itex]^{2}[/itex] - y[itex]^{2}[/itex]| < [itex]\delta^{2}[/itex]
so:
f(x,y) < [itex]\frac{x}{x^{2}+y^{2}}[/itex]([itex]\delta^{2}[/itex])
and I feel like I'm close but then I'm stuck! All help appreciated thanks !