Answer to Question #170335 in Real Analysis for Prathibha Rose

Question

Show that the function f:R² to R defined by f(x,y)= x²+y² is not uniformly continuous

Answer 1

Inorder to show that f is uniformly continuous, we show that given $\epsilon >0 \exist \, \delta>0 \ni |(x_1,y_1)-(x_2,y_2)| < \delta \implies |f(x_1,y_1) -f(x_2,y_2)|<\epsilon$

Suppose that this is true and consider two points in $\mathbb{R^2}$ say (a,b) and $(a + \frac{\delta}{2}, b + \frac{\delta}{2})$

Now consider

$|f(a + \frac{\delta}{2}, b + \frac{\delta}{2})-f(a,b)| = | (a + \frac{\delta}{2})^2 +(b+ \frac{\delta}{2})^2 - (a^2+b^2)| = |(a+b) \delta + \frac{\delta^2}{2}|$

And as $a+b$ gets larger, the quantity above is greater than $\epsilon$ which is sufficiently small.

So, we have arrived at a contradiction. Hence, we can conclude that f is not uniformly continuous.