# Thread: Proof of continuity in two dimensions

1. ## Proof of continuity in two dimensions

Firstly, i'm sorry if this is the wrong forum. It seems to be analysis, but it is in a calculus book so wasn't sure.
I'm stuck on this question:
Prove that if $g:\mathbb{R} \rightarrow \mathbb{R}$ is continuous at $a$ then $f(x,y)=g(x)$ is continuous at (a,b) for all $b \in \mathbb{R}$

If someone could just give me an idea of how to start the proof. I understand the definition for continuity.

2. Originally Posted by worc3247
\mapsto Firstly, i'm sorry if this is the wrong forum. It seems to be analysis, but it is in a calculus book so wasn't sure.
I'm stuck on this question:
Prove that if g:\mathbb{R} \rightarrow \mathbb{R} is continuous at a then f(x,y)=g(x) is continuous at (a,b) for all b \in \mathbb{R}

If someone could just give me an idea of how to start the proof. I understand the definition for continuity.
So, you are given that g is continuous. This means that for any epsilon>0, there exists a delta>0, such that |a-x|<delta implies |g(a)-g(x)|<epsilon. You want to prove a similar implication for f, that is, for a given epsilon>0, you need the existence of a delta>0, such that |a-x|<delta implies |f(a,b)-f(x,b)|<epsilon (where b is arbitrary).

However, by definition of f, we have that |f(a,b)-f(x,b)| = |g(a)-g(x)|. Let me know, if you can see where this goes.