Chapter 13 - Section 13.2 - Limits and Continuity in Higher Dimensions - Exercises - Page 693: 69

$|f(x,y) -f(0,0) | \lt \epsilon $

We have $f(x,y)=x^2+y^2 ; \\ f(0,0)=0$ Now, $|f(x,y) -f(0,0) | \lt \epsilon $ $\implies |x^2+y^2-0| \lt 0.01 $ $\implies \sqrt {x^2+y^2 } \lt 0.1$ Consider $\delta =0.1$ Now, $\sqrt {x^2+y^2 } \lt \delta$ So, $|f(x,y) -f(0,0) | \lt \epsilon $