Continuity and One- Sided Limits (1.4) September 26th, 2012.

Continuity and One-Continuity and One-Sided Limits (1.4)Sided Limits (1.4)

Continuity and One-Continuity and One-Sided Limits (1.4)Sided Limits (1.4)

September 26th, 2012September 26th, 2012September 26th, 2012September 26th, 2012

I. Continuity at a Point on an I. Continuity at a Point on an Open IntervalOpen IntervalI. Continuity at a Point on an I. Continuity at a Point on an Open IntervalOpen Interval

DefDef: A function is : A function is continuous at a point ccontinuous at a point c if if1. f(c) is defined1. f(c) is defined2. exists, and2. exists, and3. .3. .

limx→ c

f(x)

limx→ c

f(x) = f(c)

A function is A function is continuous on an open interval continuous on an open interval (a, b)(a, b) if it is continuous at each point in the if it is continuous at each point in the interval. A function that is continuous on interval. A function that is continuous on the entire real line the entire real line is is everywhere continuous.everywhere continuous.(−∞,∞)

If a function f is continuous on the open If a function f is continuous on the open interval (a, b) except at point c, it is said to interval (a, b) except at point c, it is said to have a have a discontinuitydiscontinuity at c. This discontinuity at c. This discontinuity is is removableremovable if f can be made continuous if f can be made continuous by appropriately defining (or redefining) by appropriately defining (or redefining) f(c). Otherwise, the discontinuity is f(c). Otherwise, the discontinuity is nonremovable.nonremovable.

f(c) is not f(c) is not defineddefined

Removable Removable discontinuitdiscontinuityyat c.at c.

does not existdoes not exist

limx→ c

f(x)

NonremovabNonremovable le discontinuity discontinuity at c.at c.

limx→ c

f(x) ≠ f(c)

Removable Removable discontinuity discontinuity at c.at c.

Ex. 1Ex. 1: Discuss the continuity of each : Discuss the continuity of each function.function.(a) (a)

(b) (b)

(c)(c)

(d)(d)

f (x)=2

x+5

g(x)=x2 +2x+1

x+1

h(x)=x2 −3,x≤02x−3,x> 0

⎧⎨⎩

k(x)=2cos3x

II. One-Sided Limits and II. One-Sided Limits and Continuity on a Closed IntervalContinuity on a Closed IntervalII. One-Sided Limits and II. One-Sided Limits and Continuity on a Closed IntervalContinuity on a Closed IntervalOne-Sided Limits One-Sided Limits are denoted by the are denoted by the

following.following. means the limit as x means the limit as x approaches approaches c from the right, andc from the right, and

means the limit as x means the limit as x approachesapproaches c from the left.c from the left.

limx→ c+

f(x)

limx→ c−

f(x)

Ex. 2Ex. 2: Find the limit (if it exists). If it does : Find the limit (if it exists). If it does not exist, explain why.not exist, explain why.(a)(a)

(b)(b)

(c)(c)

limx→ −3−

x

x2 −9

limx→1+

[[x−2]] +1

limx→1−

[[x−2]] +1

You try:You try: (a)(a)

(b)(b)

limx→ 5+

x−5x2 −25

limx→ 0−

xx

Thm. 1.10: The Existence of a Limit:Thm. 1.10: The Existence of a Limit: Let f Let f be a function and let c and L be real be a function and let c and L be real numbers. The limit of f(x) as x approaches c numbers. The limit of f(x) as x approaches c is L if and only ifis L if and only if

andand

limx→ c−

f(x) =L

limx→ c+

f(x) =L

DefDef: A function f is : A function f is continuous on a closed continuous on a closed interval [a, b]interval [a, b] if it is continuous on the open if it is continuous on the open interval (a, b) and and interval (a, b) and and . .We say that f is We say that f is continuous from the right continuous from the right of aof a and and continuous from the left of b.continuous from the left of b.

limx→ a+

f(x) = f(a) limx→ b−

f(x) = f(b)

Ex. 3Ex. 3: Discuss the continuity of the : Discuss the continuity of the functionfunction on the closed interval [-on the closed interval [-1, 2].1, 2].g(x)=

1x2 −4

III. Properties of ContinuityIII. Properties of ContinuityIII. Properties of ContinuityIII. Properties of Continuity

Thm. 1.11: Properties of Continuity:Thm. 1.11: Properties of Continuity: If b is a If b is a real number and f and g are continuous at real number and f and g are continuous at x=c, then the following functions are also x=c, then the following functions are also continuous at c.continuous at c.1. Scalar multiple: bf1. Scalar multiple: bf2. Sum and difference: 2. Sum and difference: 3. Product: fg3. Product: fg4. Quotient: 4. Quotient:

f ±g

f

g,g(c)≠0

Functions that are Continuous at Every Functions that are Continuous at Every Point in their Domain:Point in their Domain:1. Polynomial functions1. Polynomial functions

2. Rational functions2. Rational functions

3. Radical functions3. Radical functions

4. Trigonometric functions4. Trigonometric functions

p(x)=anxn +an−1x

n−1 +...+a1x+a0

r(x)=p(x)q(x)

,q(x) ≠0

f (x)= xn

sin x,cos x, tan x,csc x,sec x,cot x

Thm. 1.12: Continuity of a Composite Thm. 1.12: Continuity of a Composite Function:Function: If g is continuous at c and f is If g is continuous at c and f is continuous at g(c), then the composite continuous at g(c), then the composite function given by function given by is continuous at c.is continuous at c.

( f og)(x)= f(g(x))

Ex. 4:Ex. 4: Find the x-values (if any) at which f is Find the x-values (if any) at which f is not continuous. Which of the discontinuities not continuous. Which of the discontinuities are removable?are removable?

(a)(a)

(b)(b)

f (x)=3x−cosx

f (x)=21x

⎛⎝⎜

⎞⎠⎟

2

−41x

⎛⎝⎜

⎞⎠⎟+ 3

IV. The Intermediate Value IV. The Intermediate Value TheoremTheoremIV. The Intermediate Value IV. The Intermediate Value TheoremTheorem

***Thm. 1.13: The Intermediate Value ***Thm. 1.13: The Intermediate Value Theorem:Theorem: If f is continuous on the closed If f is continuous on the closed interval [a, b] and k is any number interval [a, b] and k is any number between f(a) and f(b), then there is at least between f(a) and f(b), then there is at least one number c in [a, b] such that f(c)=k.one number c in [a, b] such that f(c)=k.

Ex. 5Ex. 5: Explain why the function : Explain why the function has a zero in the interval .has a zero in the interval .

f (x)=x2 −2−cosx0,π[ ]