- #1

julypraise

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## Homework Statement

Conjecture. Suppose [itex]a\in \mathbb{R}[/itex]. Suppose [itex]f[/itex] is a real-valued function defined on [itex][a,a]=\{a\}[/itex]. Suppose [itex]x\in [a,a][/itex]. Then there exists a function [itex]\phi[/itex] defined by [itex]{\displaystyle \phi(t)=\frac{f(t)-f(x)}{t-x}\quad(a<t<a,t\neq x)}[/itex].

(i) Before proving (or disproving this) does this conjecture make sense in the first place?

(ii) If make sense, does it truly exist?

## Homework Equations

Relevant posts are:

https://www.physicsforums.com/showthread.php?t=585386

https://www.physicsforums.com/showthread.php?t=338366

## The Attempt at a Solution

(i) If I kinda restate this conjecture, it becomes:

Conjecture. Suppose [itex]a\in \mathbb{R}[/itex]. Suppose [itex]f[/itex] is a real-valued function defined on [itex][a,a]=\{a\}[/itex]. Suppose [itex]x\in [a,a][/itex]. Then there exists a function [itex]{\displaystyle \phi:\{t\in \mathbb{R}: a<t<a\} \to \mathbb{R} : t \mapsto \frac{f(t)-f(x)}{t-x}}[/itex].

So it seems make sense in the ground of first order language and ZFC. Isn't it?

(ii) I think this function is simply [itex]\emptyset[/itex] because the domain is empty set.

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