MAT 3749 Introduction to Analysis

MAT 3749Introduction to Analysis Section 2.2 Part II Intermediate Value Theorem http://myhome.spu.edu/lauw

Preview • Prove Intermediate Value Theorem (statement 2). • The first proof with substantial length • As usual, we will go through the analysis carefully • I will have questions to ask you. Please try to answer some questions.

References • Section 2.2

IVT Statement 1 • Suppose f is continuous on [a,b] with f(a)≠f(b) and N is between f(a) and f(b) • Then there is a no. c in (a,b) such that f(c)=N

IVT Statement 2 • Suppose f is continuous on [a,b] and that f(a), f(b) are with different signs • Then there is a no. c in (a,b) such that f(c)=0

Intermediate Value Theorem • There are usually two type of proofs. • Use sequences • Use contradictions to argue that and

Recall Continuity (e-d)

Continuous from the Right

Recall Supremum

The Completeness Axiom

Analysis • Suppose f is continuous on [a,b] and that f(a), f(b) are with different signs • Then there is a no. c in (a,b) such that f(c)=0

Analysis: Step 1 Without

Analysis: Step 2 Without

Analysis: Step 2

Analysis: Step 4 Why does ? Why does ?

Analysis: Step 5 What’s wrong?

Analysis: Step 5 Why is not an upper bound of ?

Analysis: Step 5 What is the consequence if is not an upper bound of ?

Analysis: Step 5 Why

Analysis: Step 5 If so, what is the contradiction?

Analysis: Step 6 Can ? Why?

Lemma 1

Lemma 2

Proof • Suppose f is continuous on [a,b] and that f(a), f(b) are with different signs • Then there is a no. c in (a,b) such that f(c)=0

Proof Without

Proof

MAT 3749 Introduction to Analysis