Here's a cute exam problem or two:
if $F\from A \to \bR$ is a function, possibly continuous/discontinuous
at any particular point, and $g\from A \to \bR$ is continuous
(IE, continuous at every point), then show that $H=F+g$ is continuous
at exactly the points where $F$ is continuous

\hint{First, show this is equivalent to: if $g$ is continuous at $x$,
 then $H=F+g$ is continuous at $x$ iff $H$ is continuous at $x$
 (IE, if you prove this for every point, you prove it for the whole domain).
 Second, this statement can be proven by contraposition, but it's slicker
 and clearer to write $F=H-g$, and use that.}

(NB: did I assign this as homework?)