Question

Let D = {(x, y) ∈ R

2

: x > 0, 0 < y < x3}. Define

f(x, y) = (

0, (x, y) ∈/ D

1, (x, y) ∈ D.

a) Approaching (0, 0) along the line y = mx for each real number m and the y-axis, prove that

lim(x,y)→(0,0) f(x, y) exists and compute the limit.

b) Argue whether f is continuous at (0, 0).

Answer 1

(a)

$\lim\limits_{(x,y)\rightarrow (0,0)} f(x,y)\\ =\lim\limits_{(x,y)\rightarrow (0,0)} f(x,mx)\\ So, \lim\limits_{(x,y)\rightarrow (0,0)} f(x,0)\leq\lim\limits_{(x,y)\rightarrow (0,0)} f(x,y)\leq\lim\limits_{(x,y)\rightarrow (0,0)} (mx)^3\\ \Rightarrow 0\leq\lim\limits_{(x,y)\rightarrow (0,0)} f(x,y)\leq0\\ Hence, \lim\limits_{(x,y)\rightarrow (0,0)} f(x,y)=0$

(b) Yes, the function is continuous at $x=0.$