Question

Let D is a subset of R² ,S_r(x₀,y₀) subset of D for some r>0 and f:D to R .prove that f is continuous at (x₀,y₀) if and only if if the limit of f as (x,y) tends to (x₀,y₀) exists and is equal to f(x₀,y₀).

Answer 1

We have given that,

D is a subset of R² ,S_r(x₀,y₀) subset of D for some r>0 and f:D to R

then,

f is continuous at $(x_o,y_o)$ if

$lim(x,y)\rightarrow(x_o,y_o)f(x,y) = f(x_o,y_o).$

This means three things:

1.) f is defined at $(x_o,y_o)$

2.) $lim(x,y)\rightarrow(x_o,y_o) f(x,y)$ exists.

3.) They are equal.

Hence according to the given conditions these three conditions are satisfied hence f is continuous.