Is the the sum of two locally bounded functions also locally bounded? Does someone know if the function resulting of the sum of two locally bounded functions is also locally bounded?
Thanks in advance!
 A: The answer is yes, and it is straightforward enough for you to try to work out for yourself.
Suppose that $X$ is a metric space and $f,g: X \rightarrow \mathbb{R}$ are two locally bounded functions.  This means that for each $x \in X$, there exists an open set 
$U_f$ containing $x$ on which $f$ is bounded and an open set $U_g$ containing $x$ on which $g$ is bounded.  We want to use $U_f$ and $U_g$ to construct an open set on which both $f$ and $g$ are bounded, for then $f+g$ will be bounded as well.
Can you think of how to do this?
A: A function $f$ is locally bounded if for every $x$ there exists an open neighborhood $U_x$ of $x$, and a constant $M_x$ (which may depend on $x$) such that $|f(y)|\leq M_x$ for all $y\in U_x$.
Suppose $f$ and $g$ are both locally bounded. Let $x$ be an element of the domain. Then there are open neighborhoods $U_x$ and $V_x$ of $x$, and constants $M_x$ and $N_x$ such that $|f(y)|\leq M_x$ for all $y\in U_x$, and $|g(z)|\leq N_x$ for all $z\in V_x$.
What can you say about $f(y)+g(y)$ if $y\in U_x\cap V_x$?
