Monday, 30 June 2014

calculus - Is f(x,y)=fracsinsqrt[3]x3+y3sqrt[5]x5+y5 uniformly continuous or not

Find out if function f(x,y)=sin3x3+y35x5+y5

is uniformly continous or not in area $D=\{0I found out that we have no limx,y0f(x,y), because limx,y0sin3x3+y35x5+y5=limρ03ρ3sin3α+ρ3cos3α5ρ5sin5α+ρ5cos5α=limρ03sin3α+cos3α5sin5α+cos5α

hence we can't use The Uniform Continuity Theorem as we can't determ f(0,0). Function doesn't have bounded partial derivatives, so I think it's not uniformly continous, but I don't know how to show that

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