1.4 Continuity f is continuous at a if

1. is defined.

2. exists.

3.

xfax

lim

af

afxfax

lim

Ex 1: Discontinuous where & why?

*see graph.

1.4 Continuity 3 types of discontinuity:

Removable Infinite Jump

a)

b)

2

22

x

xxxf

xxf

Ex 2: Discontinuous where & why?

c)

d)

2

1

xxf

2 if 1

2 if 1

x

xxxf

Ex 2: Discontinuous where & why?

Functions are continuous at every

number in their domains!

f is continuous on [a,b] if it is continuous on (a, b) and:

Continuity on a Closed Interval

afxfax

lim

bfxfbx

lim

Ex 3: Show that f(x) is continuous on the interval

[1, 1]

211 xxf

Ex 4: Continuous where?

xxf cos1ln

The Intermediate Value Theorem (IVT):

If f is continuous on the interval [a, b] and k is any

number between f(a) & f(b), then there exists a number c in (a, b) such that f(c) = k.

Ex 5: Show that the equation has a root in the

interval [1, 2]

02364 23 xxx

1.4 pg. 78pg. 781 – 5 odds,1 – 5 odds,

7 – 23 EOO,7 – 23 EOO,25 – 31 odds,25 – 31 odds,33 – 53 EOO,33 – 53 EOO,

57, 59, 75, 77, 8557, 59, 75, 77, 8523 Total23 Total