For example,
set $A = \left \{ 1, 5, 10, 30 \right \} \in \mathbb{R}^N$
r = $10$
How do I write down a set which takes $A - r$ as lower bound, and $A + r$ as upper bound, containing ranges?
like:
set $C = (A - r)?(A+r) = \left \{ [-9,11], [-5,15], [0,20], [20, 40] \right \} \in \mathbb{R}^?$
Are there any notation that could describe such $?$ operation?
If I want to know if a number is in $C$
Is it okay to use, for example $(A-r)<m<(A+r)$?
On
If I read the question correctly, you start with a point $A$ in $\mathbb{R}^N$. In your example $N=4$. $A$ is a point in that space, not a set.
What you want is the "cube" in $\mathbb{R}^N$ (same $N$) consisting of all the points each coordinate of which differs from the corresponding entry in $A$ by at most $r$.
In your example you could describe that as the product $$ [−9,11] \times [−5,15] \times [0,20] \times [20,40]. $$
Note: my answer isn't the same as @saulspatz . I think we have interpreted your question differently.
I would write $C=\{[a-r,a+r]\mid a\in A\}$ When you say, "I want to know if a number is in $C$," I think you must mean "I want to know if some number is in an element of $C$." The elements of $C$ are intervals not numbers, so it really doesn't make sense to ask if a number is in $C$.
There are (at least) two ways to express this. $$\exists c\in C(m\in c)\\ \text{ or }\\ m\in \bigcup_{c\in C} {c}$$ The question marks are not usually used in math, except on the blackboard. You would write, "Is it true that...?"