How is Epsilon Defined in the Proof of the Chain Rule in James Stewart Calculus?

[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 + \Deltax , we defined the increment of y as

\Deltay = f(a + \Deltax) - 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