- #1

julypraise

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## Homework Statement

Suppose [itex]f[/itex] is a real-valued function defined on [itex][a,a]=\{a\}[/itex].

(i) Then can we use the definition below to this function?

(ii) If it can be used, then is [itex]f'(a)[/itex] is undefined?

(iii) If it is undefined, why is it that? Is it because the [itex]\phi[/itex] function was undefinable in the first place?

## Homework Equations

Definition (Rudin, p. 103). Let [itex]f[/itex] be defined (and real-valued) on [itex][a,b][/itex]. For any [itex]x\in[a,b][/itex], form the quotient [itex]{\displaystyle \phi(t)=\frac{f(t)-f(x)}{t-x}\quad(a<t<b,t\neq x)}[/itex], and define [itex]f'(x)=\lim_{t\to x}\phi(t)[/itex], provided this limit exists (in [itex]\mathbb{R}[/itex]).

## The Attempt at a Solution

(i) It seems so, but only the author knows the truth: I don't know whether he (Rudin) allows me to use this kind of interval for the definition. (When he proposes the definition of segment and interval by notation [itex](a,b), [a,b][/itex] in the preceding pages, he does not specify what [itex]a,b[/itex] are.)

(ii) Undefined.

(iii) And the reason is seemingly that I cannot define [itex]\phi[/itex] as stated in the definition. Otherwise, no idea.

Note: I just feel so confused about these notions of 'define', 'form', 'construct', such and such because I can't translate properly these things to the first order language (logic). Please help me.