In the following article, a proof for the inequality of a convex/concave function is given. In the proof, they define $h_{a,b}$ as the segment joining two points in the convex/concave function. They state that $h_{a,b}$ is a linear function, and, accordingly, they proceed to use the two properties of a linear function namely: $$f(x+y)=f(x)+f(y)$$ $$f(ax)=af(x)$$
However, how can they be certain that the line that extends through the $h_{a,b}$ segment is a linear function and not an affine function (in the latter case it wouldn't make the two properties usable since we would have $f(0)\neq 0$)
They are defining an affine function $h$ on the line joining the points $a,b$ such that $h(a) = f(a)$, $h(b) = f(b)$.
Since $h$ is affine, it satisfies $h(tx+(1-t)y) = th(x)+(1-t) h(y)$ for all $x,y$ on the line $a,b$ and all $t$.
In particular, if we pick $x=a,y=b$ we have $h(ta+(1-t)b) = th(a)+(1-t)h(b)$. Since we have $h(a) = f(a)$, $h(b) = f(b)$, we must have $h(ta+(1-t)b) = tf(a)+(1-t)f(b)$.