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evinda

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We are given a continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f \left( \frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}, \forall x, y \in \mathbb{R}$. I want to show that $f$ is convex.I have tried the following:

Let $\lambda \in [0,1]$.

We have that $f(\lambda x+(1-\lambda)y)=f \left( \frac{2 \lambda x+ 2(1-\lambda)y}{2}\right) \leq \frac{f(2 \lambda x)+f(2(1-\lambda)y)}{2}$.But can we show that the latter is $\leq \lambda f(x)+(1-\lambda)f(y)$ ?Or can't we show it by showing that the definition of convexity is satisfied?

If not, how else can we prove the desired result? (Thinking)