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I need to know how to do this to find out the derivative of a function as close as possible to an undefined point.

Assume that $f$ is not differentiable at $x=a$.

This might be the same as asking how to evaluate

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

Not sure.

Basically, what's the derivative of a function $f$ at the point closest to $x=a$ if $f$ is not defined at $a$? Note I'm not asking for the derivative of $f(a)$, since that is undefined, but instead the point closest to the undefined point, the endpoint I suppose.

What's the endpoint and what's the derivative of it?

I'm guessing I have to break this into a piecewise function but I'd rather not.

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    It depends very much on the context. What is $f$? If you can find an expression for $f'(x)=\lim\limits_{h\to0}\frac{1}{h}(f(x+h)-f(x))$ for $x$ sufficiently close to $a$, then it is just a matter of evaluating the limit of the function $f'$ at $a$, provided these limits exist.2012-01-06
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    if f is given function ,then in brackets statement is equivalence f',so it will be limit of f' at point x=a2012-01-06
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    There is no point different from $a$ and closest to $a$.2012-01-06
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    Why not? I feel like I'm getting into infinitesimal territory and that's exactly what I'm trying to avoid.2012-01-06
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    @Korgan Rivera: Undivisible infinitesimal theory is what you are getting into. I should have said there is no real number closest to $a$ and different from $a$. **Suppose** to the contrary there is one, say $b\ne a$. Look at $|b-a|$. It is a positive real number and has the property that there is no positive real number less than $|b-a|$. But that cannot be true, $|b-a|/2$ is less than $|b-a|$. So there cannot be a real number different from $a$ but closest to $a$. (There cannot be an element $b$ of an extension of the reals either, if division by $2$ is possible there.)2012-01-06
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    I understand what you mean. I want to avoid using infinitesimals. But I can't think of a way around it.2012-01-06

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