Sec 1.7 Intro to Continuity

Idea: A continuous function has no breaks or gaps or holes.

Its graph can be drawn without lifting the pencil from the page.

Ex. is continuous on

a) Is y continuous on ?

Ex. is continuous on

a) Is f(x) continuous on ?

Ex. is continuous on

a) Is h(t) continuous on ?

b) Is h(t) continuous on ?

Ex.

p(x) is not continuous at x = 1

p(x) is continuous on

a) Is p(x) continuous on ?

b) Is p(x) continuous on ?

Ex.

r(x) is continuous on

a) Is r(x) continuous on ?

Ex.

p(x) is not continuous at x = 0

p(x) is continuous on

a) Is p(x) continuous on ?

b) Is p(x) continuous on ?

Numerical View

Intermediate Value Theorem

For f(x) continuous on [a, b] and for k a number such that

,

there must exist c so that so that