Section 2.3a

Continuity

Find the points at which the function f is continuous, and thepoints at which f is discontinuous.

In general, a continuousfunction is a function whoseoutputs vary continuously withthe inputs and do not “jump”from one value to anotherwithout taking on values inbetween.

Graphically, any function f(x) whose graph can be sketched inone continuous motion without lifting the pencil is an exampleof a continuous function…

Find the points at which the function f is continuous, and thepoints at which f is discontinuous.

The function f is continuous atevery point in its domain [0,4]except at x = 1 and x = 2. Notethe relationship between thelimit of f and the value of f ateach point of the function’sdomain.

Points at which f is continuous:

0

lim 0x

f x f

At x = 0, 4

lim 4x

f x f

At x = 4,

1,2,c At 0 < c < 4, limx cf x f c

Find the points at which the function f is continuous, and thepoints at which f is discontinuous.

The function f is continuous atevery point in its domain [0,4]except at x = 1 and x = 2. Notethe relationship between thelimit of f and the value of f ateach point of the function’sdomain.

Points at which f is discontinuous:

1

limxf x

At x = 1,

2

lim 1,x

f x

At x = 2,

DNE

1 2fbut

At c < 0, c > 4, these points are not in the domain of f

Definition: Continuity at a PointInterior Point: A function is continuous atan interior point c of its domain if

y f x

limx cf x f c

Exterior: A function is continuous at aleft endpoint a or is continuous at a right endpointb of its domain if

y f x

limx a

f x f a

or limx b

f x f b

respectively.

,

Types of Discontinuity

Removable Discontinuity – Thefunction can be redefined at a singlepoint in order to “remove” thediscontinuity (the graph has a “hole” in it).

2x x

f xx

Graph

(0,1)

Removable discontinuity at x = 0

Types of Discontinuity

Jump Discontinuity – The one-sidedlimits of the function at the given pointexist, but have different values.

1, 0

0, 0

xf x

x

Graph

(0,1)

Jump discontinuity at x = 0

Types of Discontinuity

Infinite Discontinuity – The functionapproaches positive or negative infinityas x approaches the given point.

2

1f x

xGraph

Infinite discontinuity at x = 0

Types of Discontinuity

Oscillating Discontinuity – Thefunction oscillates with increasingfrequency as x approaches the givenpoint

1sinf xx

Graph

Oscillating discontinuity at x = 0

Continuous FunctionsA function is continuous on an interval if and only if it iscontinuous at every point on the interval. A continuousfunction is one that is continuous at every point of its domain.

Ex: Consider the reciprocal function

Does this function have any points of discontinuity?

1f x

x

Is this a continuous function?

Yes The function has an infinite discontinuity at x = 0

Yes The only point of discontinuity (at x = 0) is not inthe domain of the function, so the function is continuouson its domain!!!

Theorem: Properties of Continuous FunctionsIf the functions f and g are continuous at x = c, then thefollowing combinations are continuous at x = c.

4. Constant Multiples: ,k f for any number k

5. Quotients: ,f g provided 0g c

3. Products: f g2. Differences: f g1. Sums: f g

Theorem: Composite of Continuous FunctionsIf f is continuous at c and g is continuous at f (c), then thecomposite g f is continuous at c.

Guided PracticeFor the following function, (a) find each point of discontinuity,(b) Which of the discontinuities are removable? Notremovable? Give reasons for your answers.

3 , 2

, 2

2, 2

x x

f x x x

x x

The graph?

(a) Point of discontinuity:

2x (b) Removable reassign:

2 1f