Help in proof of chain rule

[mahmoud2011]

From james stewart calculus Early Transcendentals.Before he states the proof he intoduced a property of differentiable funcion

If y=f(x) and x changes from a to a + $\Delta$x , we defined the increment of y as

$\Delta$y = f(a + $\Delta$x) - f(a)

Accordin to definition of a derivative ,we have

lim $\frac{\Delta y}{\Delta x}$ = f'(a)​

so if we denote by $\epsilon$ the difference between Difference Qutient and the derivative we obtain

lim $\epsilon$ = ( $lim \frac{\Delta y}{\Delta x}$ - f'(a) ) = 0

But $\epsilon = \frac{\Delta y}{\Delta x} - f'(a) \Rightarrow \Delta y = f'(a) \Delta x + \epsilon \Delta x$

If we Define $\epsilon$ to be 0 when $\Delta x$=0.then $\epsilon$ becomes a continuous function of $\Delta x$

My problem is how we defined $\epsilon$ to be 0 when $\Delta x$=0

where this is not in the Domain.

 

[Nanas]

what is the problem?

Like a piece wise Functions