Inorder to show that f is uniformly continuous, we show that given ϵ>0∃δ>0∋∣(x1,y1)−(x2,y2)∣<δ⟹∣f(x1,y1)−f(x2,y2)∣<ϵ
Suppose that this is true and consider two points in R2 say (a,b) and (a+2δ,b+2δ)
Now consider
∣f(a+2δ,b+2δ)−f(a,b)∣=∣(a+2δ)2+(b+2δ)2−(a2+b2)∣=∣(a+b)δ+2δ2∣
And as a+b gets larger, the quantity above is greater than ϵ which is sufficiently small.
So, we have arrived at a contradiction. Hence, we can conclude that f is not uniformly continuous.
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