The support of $f(x)= \cos(x)$

Question

The support of a function is the closure of the set of points where the function has non zero values.

The function $f(x)=\cos(x)$ is zero only at the points $x=\frac{(2k+1)\pi}{2}$, $k \in \mathbb{Z}$

So the support of $f$ is the set $\{ x \in \mathbb{R} : x \notin \frac{(2k+1)\pi}{2}$ for all $k \in \mathbb{Z} \}$

From this, how can we deduce that the support of $f$ then equal to $\mathbb{R}$?

Answer 1

So the support of $f$ is the set $\{ x \in \mathbb{R} : x \notin \frac{(2k+1)\pi}{2}$ for all $k \in \mathbb{Z} \}$

No. It is the closure of the set $\{ x \in \mathbb{R} : x \notin \frac{(2k+1)\pi}{2}$ for all $k \in \mathbb{Z} \}$.

From this, how can we deduce that the support of $f$ then equal to $\mathbb{R}$?

By showing that the closure of $\{ x \in \mathbb{R} : x \notin \frac{(2k+1)\pi}{2}$ for all $k \in \mathbb{Z} \}$ is $\mathbb{R}$.