Show bounded and convex function on $\mathbb R$ is constant

How can we show that a bounded and convex function on $\mathbb R$ is constant? Derivatives are of no use since the function does not have to differentiable. I saw an answer here I think a while ago but did not understand it at all.

Since derivatives are useless, we would have to use the definition and somehow show that the function lies between two values which are equal to each other. But I am unable to progress any further.

• This is an exact duplicate of an older question, but both the question and the answers are better here than they are there.
– 6005
Oct 7 '13 at 21:25

Suppose that $f$ is a convex function that is not constant. Then there must be $a$ and $b$ with $f(a)<f(b)$. Without loss of generality, assume that $a<b$.

Then for any $c>b$ we must have, due to convexity, $$f(c) \ge f(a) + (c-a)\frac{f(b)-f(a)}{b-a}$$ which grows without bounds as $c\to \infty$. So $f$ cannot be bounded.

• Not quite following your notation: In which interval does f lie below a straight line? Is c some x outside of the interval in which f lies above the line? Regarding the logic: You showed the implication "f convex AND f non-constant => f not bounded". How is that equivalent to "f convex AND f bounded => f constant"? Oct 7 '13 at 21:36
• The inequality is a consequence of the convexity inequality for $a$, $b$, $c$ -- that is $f(b)$ must be below or at the line between the points at $a$ and $c$. This is the same as saying that $f(c)$ must be above the line defined by the values at $a$ and $b$ (which is easiest to see geometrically, but you can also throw algebra at it). Oct 7 '13 at 21:40
• @Dat, as for logic: Yes, these are equivalent; both implications are equivalent to the statement "it is not true that $f$ is convex and non-constant and bounded", by the rule that $P\Rightarrow Q$ is the same as $\neg(P\land\neg Q)$. If in doubt, draw up a truth table. Oct 7 '13 at 21:42
• Ok I understand the origin of the expression. However I am not entirely sure of the proof: How does that "force" the function to be constant? We assumed it wasn't constant and were to able to show that it will not be bounded due to its convexity. So just one last thing before I get it: What exactly forced it to be a constant function? It would be nice to algebraically show that f gets locked in between two values which must equal each other (e.g. A <= f(b) <= B and A = B, thus f being constant). Thanks a lot! Oct 7 '13 at 22:10
• @Dat: You can also view it as an indirect proof: Let $f$ be bounded and convex. Suppose for a contradiction that it is not constant. Then (insert above argument here) it is unbounded. But this contradicts our knowledge that it is bounded. Therefore it is constant. Oct 7 '13 at 22:13

Hint: Without loss of generality, let $f(y)<f(z)$ for $y<z$. Then $$\frac{z-y}{z-x}f(x)+\frac{y-x}{z-x}f(z) \le f(y)$$ for all $x < y$. Now let $x \to -\infty$ and use boundedness of $f$.

• How'd you arrive at that expression? Convexity states that: f((1-t)a + tb) <= af(1-t) + tf(b) where a<b and 0<=t<=1. The right-hand side is a straightline. How did you go from the definition to that expression? Oct 7 '13 at 21:39
• Your "definition" is not quite correct. It should be $$f((1-t)x+tz) \ge (1-t)f(x)+tf(z).$$ Now put $t = \frac{y-x}{z-x}$. Oct 7 '13 at 21:44
• But your expression says the function lies above the line, which is not the case with convex functions, right? And that choice for t, any rationale behind that? Oct 7 '13 at 21:59
• Sorry, when I said "not quite correct" I meant $f(1-t)$ in your expression and I didn't notice that my proof is for convex upwards functions and you are asking about convex downwards ones. But the principle is the same. To understand the rationale behind that just imagine two points with different values on the graph of $f$ and try find out what can you say about infinitely distant point. Oct 7 '13 at 22:16
• Ok thanks, will do that! Oct 7 '13 at 22:24

I don't think that any of this is true: consider the function f(x) = \exp(-x) on [0,\inf). This function is strictly convex and bounded but not constant.

• "on $\mathbb{R}$"?
– mrf
Jun 8 '14 at 22:13
• Hi, every one! What happen if $f$ is a multivariable functions, i.e., $f: \mathbb R^n \to \mathbb R$? Is in this case $f$ still constant? May 9 '15 at 17:18
• See math.stackexchange.com/questions/24397/… for a generalizarion Jan 14 at 15:56