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Introductory Real Analysis

Complete question 1, 4 and 8 only.


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** Please see the attached file for the complete solution response **

(Q.1) It is said that a function (please see the attached file) is differentiable at a point a (please see the attached file) R if:
a) f is continuous at the point a
b) (please see the attached file) the limit
( 1)
does exist. In such a case, this limit is called "the derivative of f(x)" at the point a and it is denoted as f' (a).

Remark: The condition of continuity is absolutely necessary, because it is possible the limit (1) to exist, but the function not to be continuous. Example:
( 2)
Since f(x) is a constant on both sides of x = 0, we have :
(please see the attached file)
( 3)
but f is not continuous at x = 0, therefore it is not differentiable at x = 0.

It is said that a function has a limit at the ...

Solution Summary

This solution provides a detailed, step-by-step explanation of the given calculus problems.