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 $\displaystyle g:\mathbb{R} \rightarrow \mathbb{R}$ is continuous at $\displaystyle a$ then $\displaystyle f(x,y)=g(x)$ is continuous at (a,b) for all $\displaystyle b \in \mathbb{R}$

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.

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