Question

Let D₁ and D₂ be subsets of R² and let f₁:D₁ to R and f₂: D₂ to R be continuous functions such that f₁(x,y) =f₂(x,y) for all (x,y) subset of D₁ union D₂ ,let D_i = D₁ union D₂ and let f: D to R be defined by,

f(x,y) ={f₁(x,y) if (x,y) element of D₁,

f₂ (x,y) if (x,y) element of D₂

If D_i is closed for i=1,2. Prove that f is continuous

Answer 1

Let $D \subset D_1 \cup D_2$ . D is closed since it is a subset of a finite union of two closed sets, which is also closed.

Let $(x_n, y_n)$ be a convergent sequence in D. Since D is closed then $\exist (x_0,y_0) \, \in D \ni (x_n,y_n) \to (x_0,y_0).$ .

Since $D \subset D_1 \cup D_2$ we have that $f_1(x_n,y_n) = f_2(x_n,y_n)$

By the definition of f and the continuity of $f_1$ and $f_2$ , we have that f is also continuous i.e $f(x_n,y_n) \to f(x_0,y_0)$