Introduction to Mathematical Analysis I - Second Edition

LIMITS OF FUNCTIONS LIMIT THEOREMS CONTINUITY PROPERTIES OF CONTINUOUS FUNCTIONS UNIFORM CONTINUITY LIMIT SUPERIOR AND LIMIT INFERIOR OF FUNCTIONS LOWER SEMICONTINUITY AND UPPER SEMICONTINUITY 3. LIMITS AND CONTINUITY In this chapter, we extend our analysis of limit processes to functions and give the precise definition of continuous function. We derive rigorously two fundamental theorems about continuous functions: the extreme value theorem and the intermediate value theorem. 3.1 LIMITS OF FUNCTIONS Definition 3.1.1 Let f : D → R and let ¯ x be a limit point of D . We say that f has a limit at ¯ x if there exists a real number ` such that for every ε > 0, there exists δ > 0 with | f ( x ) − ` | < ε for all x ∈ D for which 0 < | x − ¯ x | < δ . In this case, we write lim x → ¯ x f ( x ) = `. Remark 3.1.1 Note that the limit point ¯ x in the definition of limit may or may not be an element of the domain D . In any case, the inequality | f ( x ) − ` | < ε need only be satisfied by elements of D . Example 3.1.1 Let f : R → R be given by f ( x ) = 5 x − 7. We prove that lim x → 2 f ( x ) = 3. Let ε > 0. First note that | f ( x ) − 2 | = | 5 x − 7 − 3 | = | 5 x − 10 | = 5 | x − 2 | . This suggests the choice δ = ε / 5. Then, if | x − 2 | < δ we have | f ( x ) − 2 | = 5 | x − 2 | < 5 δ = ε . Example 3.1.2 Let f : [ 0 , 1 ) → R be given by f ( x ) = x 2 + x . Let ¯ x = 1 and ` = 2. First note that | f ( x ) − ` | = | x 2 + x − 2 | = | x − 1 || x + 2 | and for x ∈ [ 0 , 1 ) , | x + 2 | ≤ | x | + 2 ≤ 3. Now, given ε > 0, choose δ = ε / 3. Then, if | x − 1 | < δ and x ∈ [ 0 , 1 ) , we have | f ( x ) − ` | = | x 2 + x − 2 | = | x − 1 || x + 2 | < 3 δ = 3 ε 3 = ε . This shows that lim x → 1 f ( x ) = 2.

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