MATH 109

# MATH 109 - 109 Spring 2011 Supplement on continuity of...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 109 Spring 2011 - Supplement on continuity of real-valued functions. Earlier, we defined functions in general (Eccles chapter 8). An important special case consists of functions on the reals, f : R → R . Example. • Polynomials, exponentials, trigonometric functions • Modulus function | x | = ( x if x ≥ − x if x < • Step function H ( x ) = ( i f x < 1 i f x ≥ • Floor and ceiling functions b x c = largest integer less than x d x e = smallest integer greater than x Definition. Let f : R → R and x ∈ X . Then f has a limit L at x if for each ∈ R + , there is a δ ∈ R + such that for all x ∈ R , < | x − x | < δ = ⇒ | f ( x ) − L | < . Example. The function given by f ( x ) = x 2 has a limit 0 at x = 0. Proof. Let ∈ R + . We want to find δ such that for each x with 0 < | x | < δ , 0 < | x 2 | < . Define δ = √ . Then suppose 0 < | x | < δ . | x 2 | = x 2 < δ 2 = ( √ ) 2 = . Example. The modulus function has a limit 0 at x = 0. Proof. Let ∈ R + . We want to find δ such that for each x with 0 < | x | < δ , 0 < || x |− | < . Define δ = . Then suppose 0 < | x | < δ . || x | − | = | x | < δ = . Example. The function given by f ( x ) = ( | x | x if x = 0 i f x = 0 does not have a limit at 0. Proof. Notice that if x < 0 then f ( x ) = − 1 and if x > 0 then f ( x ) = 1. We will show that for any L ∈ R , L is not the limit of f at x = 0. Let L be any number and let = 1....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

MATH 109 - 109 Spring 2011 Supplement on continuity of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online