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

[itex]\Delta[/itex]y = f(a + [itex]\Delta[/itex]x) - f(a)

Accordin to definition of a derivative ,we have

lim [itex]\frac{\Delta y}{\Delta x}[/itex] = f'(a)

so if we denote by [itex]\epsilon[/itex] the difference between Difference Qutient and the derivative we obtain

lim [itex]\epsilon[/itex] = ( [itex] lim \frac{\Delta y}{\Delta x}[/itex] - f'(a) ) = 0

But [itex]\epsilon = \frac{\Delta y}{\Delta x} - f'(a) \Rightarrow \Delta y = f'(a) \Delta x + \epsilon \Delta x[/itex]

If we Define [itex]\epsilon[/itex] to be 0 when [itex]\Delta x[/itex]=0.then [itex]\epsilon[/itex] becomes a continuous function of [itex]\Delta x[/itex]

My problem is how we defined [itex]\epsilon[/itex] to be 0 when [itex]\Delta x[/itex]=0

where this is not in the Domain.