I am looking for a **reference** for the following claim:

Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.

Suppose that "$\phi$ is convex at $c$". i.e. for any $x_1,x_2>0, \alpha \in [0,1]$ satisfying $\alpha x_1 + (1- \alpha)x_2 =c$, we have $$ \phi(c)=\phi\left(\alpha x_1 + (1- \alpha)x_2 \right) \leq \alpha \phi(x_1) + (1-\alpha)\phi(x_2) . $$

Then $\phi$ satisfies Jensen ineqaulity "at $c$".

**Finite form:**

Given $\lambda_i \in [0,1],x_i\in(0,\infty),i=1,\dots,k$ such that $\sum_{i=1}^k \lambda_i=1,\sum_{i=1}^k \lambda_ix_i=c$, we have $$\phi(\sum_{i=1}^k \lambda_ix_i) \le \sum_{i=1}^k \lambda_i \phi(x_i).$$

**A more general probabilistic (measure-theoretic) form:**

Given a random variable $X \in (a,b)$ with expectation $E(X)=c$, we have $$ \phi(c)=\phi(E(X)) \le E(\phi(X)). $$

In addition, if $\phi$ is **strictly** convex at $c$, then equality holds if and only if $X$ is constant a.e..

Both of these forms of Jensen inequality follow from the existence of a supporting line to the graph of $\phi$ at $c$.

The proof of the latter fact is not hard, but I couldn't find a source in the literature that presents this "localized" form of Jensen inequality, under the sole assumption of "convexity **at a point**". (In fact, I couldn't even find the term "convex at a point" anywhere...).

I find it impossible to believe that this doesn't show up in existing literature. Any help would be welcomed.

*Comment:*

Convexity at $c$ does not imply that the one-sided derivatives exist, so the standard proof for the existence of a supporting line (subgradient) does not apply here. (When the function is convex on an interval, every number between the two-sided derivatives form a subgradient).