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- Jul 18, 2013

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I know this is probably a dumb question, but I have a question regarding this. My textbook says the following: "A function

It then follows with and example regarding if f(x) = |x| is differentiable at x = 0. They prove this by finding the limit of its derivative, and then splits it in two equations: for the limit as h -> 0

My question is as follows: does the method, as demonstrated above, work for all functions when you're trying to find if a certain point is differentiable? I question it because even

EDIT:

Sorry if this is confusing, and if you want me to clarify, I can provide a hypothetical situation.

Ok, time to clarify what I'm trying to convey with a

We are asked to see if f(x) = |x| is differentiable at x = 0. Let's PRETEND that the limit exists at f'(a). That when you took the limit from the positive and negative side of it, it resulted in 5. However, it is actually

That being said, would additional steps be taken/required, in order to prove if something is or is not differentiable. As we see in my hypothetical situation, just determining the limit was not enough.

*f*is differentiable at*a*if*f*'(a) exists."It then follows with and example regarding if f(x) = |x| is differentiable at x = 0. They prove this by finding the limit of its derivative, and then splits it in two equations: for the limit as h -> 0

^{+}and h -> 0^{-}. Finally concluding that it is not differentiable as the**limits are different.**Done.My question is as follows: does the method, as demonstrated above, work for all functions when you're trying to find if a certain point is differentiable? I question it because even

**if**the aforementioned limit DOES exist, it doesn't mean that**f'(a) exists**. It just means that it has a limit, as you can have a limit of something when f'(a) is undefined. Thus, making the method inadequate to making such a conclusion.EDIT:

Sorry if this is confusing, and if you want me to clarify, I can provide a hypothetical situation.

**hypothetical situation (not real)**

We are asked to see if f(x) = |x| is differentiable at x = 0. Let's PRETEND that the limit exists at f'(a). That when you took the limit from the positive and negative side of it, it resulted in 5. However, it is actually

**undefined**at 5, but the limit as you approach f'(a) is 5. By using the same logic as above, the logic that the textbook example used, we are to assume that YES, It is differentiable, BECAUSE the limit exists! However, that is actually NOT the case, since the by the definition,

*f*'(a) MUST exists. But in this example, YES, the limit exists, but f'(a) doesn't exist as it is undefined at a.

That being said, would additional steps be taken/required, in order to prove if something is or is not differentiable. As we see in my hypothetical situation, just determining the limit was not enough.

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