Lets say f is defined on an Interval $I = [a,b] $. Since f is convex, one immediately knows that f is continuous on $I^°$ , however left are the points $a$ and $b$
The piecewise continuity of f states, that f is continuous on certain open intervals and that the one sided limit of the endpoints of those interval exist.(but don't have to be the same!)
In an exercise it was postulated, that f is then also continuous at the endpoints a,b of f.
However I don't see, how this holds, if just the limits of the endpoints exist due piecewise continuous.
I think, there is an error in the definition of piecewise continuous, as long as one doesn't know that the one sided limits of f'(a), f'(b) exist, f(a) and f(b) should not have to be necessarily continuous. I am a bit unsure about this and would like to have a 2th opinion..
As always thanks in advance.