# Analysis problem

1. Dec 7, 2006

### kreil

Let X be any non-empty set, equipped with the "trural" metric:

$$P_x(x,y)= \{ 0:x=y,1:x \ne y \}$$

Describe all continuous functions f: X -> R (First describe what it means for a function f to be continuous at a point c in X).

I'm really quite lost on this, any help is appreciated.

Josh

2. Dec 7, 2006

### AKG

What do the open balls in X look like?

3. Dec 7, 2006

### HallsofIvy

Staff Emeritus
I'd never seen the term "trural" metric before. I would call that the "discrete" metric. I would also use
d(x,y) instead of Px(x,y). If x is an argument of the function, what does the subscript x mean?

As AKG suggested, what do open balls look like? More precisely, what do
{y| d(x,y)< 1/2} and {y| d(x,y)< 2} look like? What are the open sets in this metric?

I presume your definition of "continuous function" is one for which f-1(C) is open whenever C is open.

4. Dec 7, 2006

### kesh

yeah. trural threw me too. if it's the discrete (as it appears, though the notation is odd) all sets are open and the question is easy