# Introductory Real Analysis

Complete question 1, 4 and 8 only.

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Solutions:

(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.