# Epsilon Delta definition of a Derivative

calculusderivativesepsilon-deltalimitsreal-analysis

The derivative at a specific point $$c$$ is represented as a limit by:
$$f'(c) = \lim_{x\to c} \frac{f(x) – f(c)}{x – c}$$

It's clear to me that the epsilon delta definition of a derivative at a point $$c$$ would be:

$$\forall \epsilon > 0 ~\exists \delta > 0 \forall x: \\ 0 < |x-c| < \delta \rightarrow |\frac{f(x)-f(c)}{x-c} – L| < \epsilon$$

What's unclear to me is how to formally represent the derivative as a function of $$x$$, rather than only at point $$c$$. Basically, how would we represent this limit formally (the $$\Delta x$$ is the part tripping me up):

$$f'(x) = \lim_{\Delta x\to0} \frac{f(x + \Delta x) – f(x)}{\Delta x}$$

You do know the definition of limit, right? So, just apply it. We can argue that if the derivative at $$x$$ is $$L$$, then $$\forall \epsilon > 0 ~\exists \delta > 0 \;\forall \Delta x: \\ 0<|\Delta x| < \delta \implies \left|\frac{f(x + \Delta x) - f(x)}{\Delta x}-L\right| < \epsilon$$ Also, the derivative as a function, $$f^\prime (x)$$ is simply a function which takes a point in, and spits out the derivative of $$f$$ at that point. So, you can also define definition of a derivative at a point $$c$$ and collect all those derivatives under a function.