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In some calculus texts, one comes across the following statement - "A function is differentiable, if it has a derivative and a function has a derivative, if it is differentiable.". Two such texts are - A treatise on Advanced Calculus by Philip Franklin(this book can be found on archive.org - check section 69 - Page 109) and Advanced Calculus - An Introduction to Linear Analysis by Leonard F. Richardson - Chapter 4 - Page 101. Analytical proofs are also given to justify these statements.

This raises the question - What is the physical difference between being differentiable and having a derivative?

The book by Leonard Richardson gives some analytical explanation about the existence of a linear function, which approximates the function at points very close to x. If such a function exists, then the function is said to be differentiable(and then he goes on to prove analytically that the function has a derivative and vice versa). The problem is that the whole proof is anlaytic, and is not illustrated using a practical example. When we say that a function is differentiable, we generally check for the existence of its derivative (i.e the limit of its differential coefficient). But this text 'implies' that its possible to check the differentiability of a function by checking for the existence of such a linear function(whose slope as you've guessed becomes the derivative in the limit). So what essentially is the difference, between this approach and the standard approach? It seems purely algebraic to me....Or is there more to it?

A related question is - what's the difference between Caratheodory's definition of the derivative and the standard definition of the derivative?. The difference seems purely algebraic to me.I would like to know if there's more to it.(I think the answer to this question is closely related to the first, but since Caratheodory's definition is rarely covered in most standard calculus texts, I've decided to gain as much information as possible, on this subject.)

I am not a mathematics major, so there are limits to my theoretical thinking! I would greatly appreciate if practical examples are provided to illustrate these concepts.

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What is a "physical difference"? –  joriki Feb 16 '13 at 15:12
    
Possible duplicate: math.stackexchange.com/questions/23902/… Regards –  Amzoti Feb 16 '13 at 16:54
    
The question 23902 concerns comparing the derivative to a differential. The present question is about "derivative" versus "differentiable", which seems peculiar, but not really the same as asking about a comparison between derivatives and differentials. –  coffeemath Feb 16 '13 at 18:35
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The statement you've written is the definition of a function being differentiable, so it doesn't make sense to ask, "what the physical difference is". –  Vectk Feb 17 '13 at 3:38
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As a contrast, let me point you to coffeemath's, joriki's or Amzoti's comments. They've not answered my question, but they've not dismissed it as trivial. I've not reacted to them. –  Nikhil Panikkar Feb 18 '13 at 10:00
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For the second question, there is no difference between the two definitions - this is a simple exercise. For the first, there is also no difference. You've defined differentiability through the exist of a linear mapping which approximates the function near $x.$ The derivative is simply the matrix which represents this function; the proof given in Franklin's book that the two are equivalent is because he essentially defines differentiable by multiplying both sides of the limit definition of a derivative by $\Delta x,$ which goes to $0,$ so it may not be entirely clear you can do this (if you're not used to this sort of thing). Anyway, there's no difference.

For your comment above - if you define a fruit with certain characteristics as an apple, then you have a fruit which has characteristics that are easily seen to be equivalent, you need to actually prove that the characteristics are equivalent before you can conclude it's an apple. This is what they're doing - it's a trivial statement, but for the sake of a "rigorous" textbook, they've done it.

As an edit: If you have a LINEAR function that approximates the function $f$ you are looking at arbitrarily close to a point, this is the tangent line in $2$-dimensions - now, the function has a slope; this is the derivative. In higher dimensions, you will have a hyperplane that approximates this point - hyperplanes are translates of a linear mapping; linear mappings admit a matrix representation. This is your derivative.

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Ok, I understand your line of arguement. But now my question is - To know the existence of the linear function, should you not know what the derivative is, first? This question also applies to Caratheodory's definition.That's why I asked for a practical example! –  Nikhil Panikkar Feb 18 '13 at 7:09
    
Also as regards your comment on it being a trivial statement, I have a related question. In rigorous proofs of the chain rule, Carathadeory's definition is used in many textbooks (even though they might not use the name.(Courant, Smirnov, Apostol etc). If this statement is trivial, then there must be an alternate method of proof? –  Nikhil Panikkar Feb 18 '13 at 7:19
    
That books use the same method is not even a hint of an argument for the non-triviality of anything. –  Mariano Suárez-Alvarez Feb 18 '13 at 7:20
    
@lyj,I am actually at college,and have a lecture scheduled right now. So I may not be immediately able to respond. But do post your answer. Nice talking to you. I am glad that finally someone undertood my question! –  Nikhil Panikkar Feb 18 '13 at 7:22
    
@Mariano, I am not arguing that it is non-trivial. I am just asking, if it is non trivial, then there must be definitely an alternate proof. –  Nikhil Panikkar Feb 18 '13 at 7:27
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You're making a big deal out of nothing. There is no a difference. A function $f: \mathbb{R} \to \mathbb{R}$ is said to be differentiable at $a$ if the following limit exists

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

If the above limit exists, then it's called the derivative of $f$ at $a$, denoted as $f'(a)$. That's all there is to it.

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If there's no difference, then why devote a proof to it? Do I need to prove something, which I define? And what about my second question? You've avoided it completely. I think you should first go through the books, I've referred to, before dismissing it as 'nothing'. I did not find your answer helpful in any way. –  Nikhil Panikkar Feb 17 '13 at 11:20
    
To make my comment above more clear - If I define a fruit with certain characteristics as an apple, then need I prove that it is an apple? –  Nikhil Panikkar Feb 17 '13 at 12:11
    
There are many pairs of things which are the same but for which we need to prove that they are the same. Why is this surprising? –  Mariano Suárez-Alvarez Feb 18 '13 at 7:19
    
@Mariano, Its not surprising, if you are taking a particular instance, and trying to prove that it fits the general definition. But if you are preparing a general definition, and then trying to prove it generally, it doesn't make sense. lyj's arguement that the two conditions are 'equivalent' not equal makes more sense than your comment. –  Nikhil Panikkar Feb 18 '13 at 7:36
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