I have a quick question. If a continuous function $f$ is convex on $(a,b)$, then the following is true? Could you explain why or why not it is true?
$for \,\,x\in(a,b)$, $t=\frac{x}{b}<1$
Thus,
$f(ta + (1-t) b) \leq tf(a)+(1-t) f(b)$
by the convexity of $f$.
In short, can I apply the convex property on the boundaries of the domain of the function? Thanks in advance