# Thread: Metric spaces and continuous functions

1. ## Metric spaces and continuous functions

Let $\displaystyle (X,d)$ metric space. Let $\displaystyle Y=\mathcal B\mathcal C(X,\mathbb R)$ be the space of every continuous and bounded functions from $\displaystyle X$ to $\displaystyle \mathbb R.$

Prove that $\displaystyle f_1(x)=\dfrac{d(x,x_0)}{1+d(x,x_0)}$ and $\displaystyle f_2(x)=d(x,x_1)-d(x,x_0)$ belong to $\displaystyle Y.$

How to do this? By proving that $\displaystyle f_1,f_2$ are both continuous and bounded functions?

Well clearly both of them are continuous. The first one is bounded by 1 and the second one can be bounded by the triangle inequality. Is that all?

2. Originally Posted by Connected
Let $\displaystyle (X,d)$ metric space. Let $\displaystyle Y=\mathcal B\mathcal C(X,\mathbb R)$ be the space of every continuous and bounded functions from $\displaystyle X$ to $\displaystyle \mathbb R.$

Prove that $\displaystyle f_1(x)=\dfrac{d(x,x_0)}{1+d(x,x_0)}$ and $\displaystyle f_2(x)=d(x,x_1)-d(x,x_0)$ belong to $\displaystyle Y.$

How to do this? By proving that $\displaystyle f_1,f_2$ are both continuous and bounded functions?

Well clearly both of them are continuous. The first one is bounded by 1 and the second one can be bounded by the triangle inequality. Is that all?
Indeed.