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### Mid-point convexity does not imply convexity [duplicate]

A function $f: X \rightarrow \mathbb{R}$ is said to be mid-point convex if for all $x, y \in X$, we have $$f(\frac{x + y}{2}) \leq \frac{f(x) + f(y)}{2}.$$ Can you please give an example of a ...
This question comes from Rudin's book "principles of mathematical analysis" chapter 4,exercise 24,on page 101. The original question is: Assume that f is a continuous real function defined in $(a,b)... 6answers 13k views ### Midpoint-Convex and Continuous Implies Convex Given that $$f\left(\frac{x+y}{2}\right)\leqslant \frac{f(x)+f(y)}{2}~,$$ how can I show that$f$is convex. Thanks. Edit: I'm sorry for all the confusion.$f$is assumed to be continuous on an ... 3answers 5k views ### Using the definition of a concave function prove that$f(x)=4-x^2$is concave (do not use derivative). Let$D=[-2,2]$and$f:D\rightarrow \mathbb{R}$be$f(x)=4-x^2$. Sketch this function.Using the definition of a concave function prove that it is concave (do not use derivative). Attempt:$f(x)=4-x^2$... 1answer 254 views ### An explicit construction for an everywhere discontinuous real function with$F((a+b)/2)\leq(F(a)+F(b))/2$? I would like to know an explicit method on constructing an everywhere discontinuous real function$F$with the property: $$F((a+b)/2)\leq(F(a)+F(b))/2.$$ There is a non-constructive example (with the ... 0answers 950 views ### mid-point convex but not a.e. equal to a convex function I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpinski's theorem from which we can deduce that for ... 1answer 232 views ### Example of function satisfying for fixed$t\in (0,1)$inequality$f(tx+(1-t)y) \leq tf(x)+(1-t)f(y)$I would like to know an example of function$f: \mathbb{R} \rightarrow \mathbb{R}$which is not convex but satisfies for fixed$t\in (0,1)$the following inequality: $$f(tx+(1-t)y) \leq t f(x)+(1-t)f(... 1answer 120 views ### A function that is less than its average on a neighborhood of every point is convex Let f: (a,b) \to \Bbb R be a continuous function such that$$\forall x \in (a,b)\; \exists \epsilon_0 \forall \epsilon < \epsilon_0: f(x) \leq \frac{f(x+\epsilon)+f(x-\epsilon)}{2}. $$Is ... 1answer 87 views ### How to find an everywhere discontinuous real function with F((a+b)/2)<(F(a)+F(b))/2? In here I posted a non-constructive everywhere discontinuous real function with$$F((a+b)/2)=(F(a)+F(b))/2$$based on the using of Hamel basis. And Conifold answered there that there is no explicit ... 1answer 41 views ### Is it necessary that$f$is convex on$D$? Let$D$be a convex set in$\mathbb{R}^n$. Suppose there exists some$a>0, b>0$and$a+b=1$such that for any$x,y \in D$, the inequality$f(ax+by) \leq af(x)+bf(y)$holds. Is it necessary that$...
Let $f:I \rightarrow R$ be a function satisfying the equation $f(\dfrac{x+y}{2}) \leq \dfrac{f(x)+f(y)}{2}$ The question is, 1)Is $f$ continuous when $I$ is closed? 2)Is $f$ continuous when $I$ is ...