What should be the definition of a bounded function?

Question

If $X, Y$ are metric spaces and $f: D \subseteq X \longrightarrow Y$ is a function. What should be the definition of a bounded function?

My try:

The function $f$ is bounded if $\quad \exists$ $r>0 $ such that $ \forall$ $x, y\in D \quad d(f(x), y)<r$.

I don´t know if it´s correct my definition. Any suggestions would be great!

Answer 1

I would rather say $$\exists r > 0 \text{ such that } \forall x,y \in D, \text{ } d(f(x),f(y)) < r$$

Or equivalently $$ \exists y \in Y,\text{ } \exists r > 0 \text{ such that } \forall x\in D, \text{ } d(f(x),y) < r$$ Or equivalently $$\forall y \in Y, \exists r > 0 \text{ such that } \text{ } \forall x\in D, \text{ } d(f(x),y) < r$$

Answer 2

The definition you proposed is not correct.

First, it is not even well defined as $y$ belongs to $D$ the domain, while $f(x)$ belongs to the codomain. Hence $d(f(x),y)$ may not be properly defined.

Second you can use following definition: it exists $y_0 \in Y$ and $r >0$ such that for all $x \in D$, $d(f(x),y_0) \le r$.