If we have a function in plane, then a differential coefficient with respect to is to be defined as:
Now, in the expression above means "allow to approach but not equalize to 0."
Put ( is a natural number) : then
is equivalent to .
Now I will explain about . We can conceptually understand "natural numbers is an infinite set," and still it is practically impossible to "list every natural number." In other words, can be explained only philosophically. The same is the case with .
Accordingly, there was introduced technique that requires mathematical strictness and does not use "infinite".
technique uses the expression:
that has the wording "for every > 0, there exists a > 0,such that : if , then ".
Next, I will describe the meaning of technique. To describe briefly, take the following meaning of "A function is continuous":
The expression above can be illustrated as follows:
"" as in "for every " may be either =600 or =2 and typically means very small or infinitesimal values such as =0.0013 and =10-23. "there exists a " means that there is an "" range such that satisfies "" and that satisfies .
For example, Function is discontinuous at a point as shown in the following figure:
For a substantially small there exists no such that satisfies near in range , so we cannot define .
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