In my real analysis book i am given definition and theorem.
Definition.
Real function $f$ is convex on $I\subseteq \mathbb{R}$ if $(\forall x_1,x_2 \in I)$ $f(\frac{x_1+x_2}{2})\leq\frac{f(x_1)+f(x_2)}{2}.$
Theorem(Jensen's inequality).
Let $f:I\rightarrow\mathbb{R}$ be function defined on open interval $I\subseteq \mathbb{R}$. Function $f$ is continious and convex on $I$ if and only if given inequality is satisfied
$(\forall x_1,x_2\in I)$,$(\forall t\in [ 0,1 ] )$, $f((1-t)x_1+tx_2)\leq(1-t)f(x_1)+tf(x_2)$ $(4.29)$
If two sides in $(4.29)$ are equal for some $t\in \langle0,1\rangle$, then they are equal $\forall t\in [0,1]$
Question:
Isn't it enough for function to be convex that Jensen's inequality is satisfied for $t=\frac{1}{2}$. I am just reading the definition.