Berkeley pointed out that presuming a number as small and then afterwards accepting it as zero is illogical in finding a derivative.moreover, why would An average slope gets closer and closer to the Instantaneous rate as delta x gets smaller and smaller?does the instantaneous rate change gradually Or discontinues? If so what proof that instantaneous rate change continually?

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In the def'n of the derivative, we are not presuming a number as small and then accepting it as 0; we are simply asking what a different quotient approaches if we let deltax approach (never attain) 0.

One may accept the instantaneous rate of change as the limit of the average rate of change as definitional. Think about it thusly: Suppose I give you two points in time and the positions you are at those times; then you can compute the average velocity for that interval. Suppose I took a snapshot of you running at a particular time in that interval and I ask: at the time I took the picture, what is your velocity? It may seem a strange question, but it makes sense that there is a single associated velocity (given that there is movement) at that particular time (assuming certain "nice" properties of your position function). The way one can quantify the magnitude of this movement (i.e. velocity in particular) is by taking the limit of the difference quotient over an arbitrary time interval and letting the time interval approach 0.

If I am understanding your question, you are asking "does the instantaneous rate of change change gradually?" or to rephrase (if I understand what you are getting at) "Is the derivative function continuous?" If this is the question, the answer is: It depends. The continuity of the derivative function depends on the function whose derivative we are finding; for instance, the absolute value function's derivative is NOT continuous.

For proof of the above example (i.e. discontinuity of the absolute value's derivative), it may be helpful to look into real analysis (for a start, elementary def'ns on continuity and such) in order to not get too technical here.

Hope this helps.

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