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May you check if my proof is correct? Thank you so much! Happy Thanksgiving!

Proof.

Let assume for the search of contradiction that f'(a) is not unique.

Then f'(a)= C, and f'(a)= D, with C not equal to D.

By the definition of differentiability.

Differentiability Definition

Then C would be equal to that limit, and D would be also equal to that limit.

When a limit exists, it is unique. Otherwise, we say that the limit does not exist.

Then C=D.

Therefore, it f is differentiable at x=a, f'(a) is unique. Q.E.D.

Beginner
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    I think you can simplify it with a direct proof just by saying the derivative is a limit and limits are unique.. but if I were you I would first make you're allowed to use that fact since it kind of trivializes the problem. –  Nov 23 '16 at 23:00
  • Thank you so much for your helpful and very interesting comment! – Beginner Nov 23 '16 at 23:07

1 Answers1

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I think what you are trying to say is that you want to show $f'$ is the unique map such that:

$$\lim_{h \to 0} \frac{f(a+h) - f(a) - f'(a)h}{h} = 0$$

Hence, what you need to show is that if $g$ is any other map with this property i.e:

$$\lim_{h \to 0} \frac{f(a+h) - f(a) - g'(a)h}{h} = 0$$

then necessarily $f'=g'$.

Faraad Armwood
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