Example of a function
- Tolaso J Kos
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Example of a function
Find a function $f$ defined in $\mathbb{R}$ that is not constant and in every interval $(x_1, x_2)$ there exists an $a$ such that:
$$f(a) \geq \max \{ f(x_1), \; f(x_2) \}$$
$$f(a) \geq \max \{ f(x_1), \; f(x_2) \}$$
Imagination is much more important than knowledge.
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Re: Example of a function
The function which is equal to $1$ everywhere except at $0$ on which it is equal to $0$ is such a function.
- Tolaso J Kos
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Re: Example of a function
How about the non continuous functions of the Cauchy equation?
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Re: Example of a function
Why try something complicated while something simple would do?
Maybe you wanted to demand a strict inequality? In that case my simple example does not work any more but the non-continuous solutions of Cauchy's equation do work. However there are still simpler functions which also work. For example
\[ f(x) = \begin{cases} 0 & \text{if $x=0$ or if $x$ is irrational} \\
q &\text{if $x\neq 0$ is rational with $x=p/q$ where $q>0$ and $\mathrm{gcd}(p,q)=1$}\end{cases}\]
Maybe you wanted to demand a strict inequality? In that case my simple example does not work any more but the non-continuous solutions of Cauchy's equation do work. However there are still simpler functions which also work. For example
\[ f(x) = \begin{cases} 0 & \text{if $x=0$ or if $x$ is irrational} \\
q &\text{if $x\neq 0$ is rational with $x=p/q$ where $q>0$ and $\mathrm{gcd}(p,q)=1$}\end{cases}\]
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