Limit in the definition of derivatives is unique.

Question

We know by first principle derivatives of function f at 'a' is given by f'(a)=limh→0[(f(a+h)-f(a))/h] exists...........(1) From our basic calculas it is also known that limit is unique.we also know that in a non Hausdorff space limit of a function need not be unique ,now my question is if f is differentiable then whether limit at right hand side is going to be unique no matter which topology we take. Thanks in advance

Answer 1

Derivatives make no sense in arbitrary topological spaces, so we are in no danger of running into that problem. Say $f\colon X\rightarrow Y$ is a map between topological spaces. Then what does the quotient $$\frac{f(a+h) - f(a)}{h}$$ even mean? You are adding elements of $X$. This operation is not available in some arbitrary topological space. Then you subtract $f(a)$ from $f(a+h)$ and divide by $h$. Again, these operations are not available in the general context of topological spaces. Note also that $h$ appears both as in conjunction with elements of $X$ and elements in $Y$, so wherever $h$ comes from, this set must operate on both $X$ and $Y$. To define the derivative you must impose some sort of algebraic structure on $X$ and $Y$. Moreover, for the derivative to be meaningful, the algebraic structure on $X$ and $Y$ must also be meaningful. This brings us all the way to Euclidean spaces and their more immediate generalizations (e.g. normed linear spaces).