Equivalent definitions of support

Question

Let $X\in\mathbb R^m$ be a set and $f:X\to \mathbb R^n$ be a function. The book I'm using define the support of this map as:

$$\operatorname{supp}f=\{x\in X\mid\text{$x=\lim x_k$ with $x_k\in X$ and $f(x_k)\neq$ 0 for every $k\in \mathbb N$}\}$$

I would like to know if this definition is equivalent to the definition below:

The support of a function is the closure of the set of the points $x\in\mathbb R^m$ such that $f(x)\neq 0$

Answer 1

Yes it's equivalent if you have normal definitions for the other concepts.

How you prove it depends on the definition of closure you're using, but one useful definition in this case is that the closure of a set $M$ is defined as:

$$\bar M = \{x \in X | x = \lim x_n, x_n \in X \}$$

That way it becomes quite clear that the first definition talks about the closure of the set where $f(x)\ne 0$.

If you're not using that definition of closure it should be quite straight forward to prove that your definition is equivalent with the above.