Linked Questions

4
votes
1answer
1k views

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 ...
3
votes
1answer
60 views

if there exists a discontinuous function f(x) which satisfies $f(\frac{x+y}{2})\leqslant\frac{1}{2}f(x)+\frac{1}{2}f(y)$ but is not convex? [duplicate]

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)...
30
votes
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 ...
4
votes
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: $f(x)=4-x^2$ ...
7
votes
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 ...
4
votes
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 ...
5
votes
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(...
5
votes
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 $...
2
votes
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 ...
0
votes
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 $...
1
vote
1answer
26 views

condition of convexity when midpoint convex

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 ...