continuity - Find all continuous functions that satisfy the Jensen inequality(?) $f(frac{x+y}{2})=frac{f(x)+f(y)}{2}$

Find all continuous functions that satisfy the Jensen inequality(I don't know why the problems states ''inequality'') $$f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$$

I've checked around stackexchange and google a bit, but none of the questions related to this problem gave me any hints on what I should do. The expression ''find all continuous functions'' confuses me. Am I supposed to determine some kind of a property that applies to all continuous functions that satisfy the property above or?

Likewise there is a problem asking to find all continuous functions that satisfy$$f(x+y)=f(x)+f(y)$$
I'm sorry if this is a duplicate question, but even after reading all of those answers I've failed to find an adequate one. Also, another problem is that I'm not allowed to use derivations, and hence anything more complex than that.

Answer

The first question is actually similar to the second question $$f\left(\frac{x+y}{2} \right)=\frac{f(x)+f(y)}{2}$$
Then we set $f(x)-f(0)=g(x)$. Note that $g(x)$ is also a solution to the functional equation. Now, let $x=2a, y=0$. Then notice $$2g(a)=g(2a) \implies g(a)=2g\left(\frac{a}{2}\right)$$
So $$g(x)+g(y)=2g\left(\frac{x+y}{2} \right)=g(x+y)$$
So the solutions to the first question are functions that satisfy the second question, plus some arbitrary constant $c$.

The second question is called the Cauchy Functional Equation, answered here.