TITLE: If $f>0$ is discontinuous and bounded on $I$, is there a $g>0$ bounded such that $fg$ is continuous at least at one point?
QUESTION [5 upvotes]: Let $I\subseteq\mathbb{R}$ be a closed and bounded interval and let
$$f:I\to(0,\infty)$$
be a bounded function that is discontinuous at every point in $I$. 
Does there exist a bounded function $g:I\to(0,\infty)$ such that the product $fg$ is continuous at least at one point in $I$?

REPLY [2 votes]: Not necessarily.  Let $f(x) = 1/2^n$ if $x = m/2^n$ for some $m \in \mathbb{N}$, and 1 otherwise.  If $fg$ is continuous at $x_0$, then for some $a>0$ we have $g(x) \approx a/f(x)$ near $x_0$.  Then you can show that $g$ is unbounded.  (In fact, it is unbounded in every neighborhood of $x_0$.)
Do you see how to make this argument precise?