1.4Continuity

andOne-Sided Limits

This will test

the “Limits”of yourbrain!

Definition of Continuity

A function is called continuous at c if the following three conditions are met:

1. f(c) is defined

2. existsxfcx

)(lim→

3. )()(lim cfxfcx

=→

A function is continuous on an open interval (a,b) if it is continuous at each point in the interval.

Two Types of Discontinuities

1. Removable Point Discontinuity2. Non-removable Jump and Infinite

Removable example

1

1)(

2

−−

=xx

xf

1

)1)(1()(

−+−

=xxx

xf

the open circlecan be filled into make itcontinuous

Non-removable discontinuity.

Ex. x

xx 0lim→

=−→ x

xx 0lim -1

=+→ x

xx 0lim 1

Determine whether the following functions arecontinuous on the given interval.

( )1,0,1

)(x

xf =

( )1

yes, it iscontinuous

)2,0(,1

1)(

2

−−

=xx

xf

( )

discontinuous at x = 1

removable discontinuity since filling in (1,2)would make it continuous.

)2,0(,sin)( πxxf =

π2

yes, it is continuous

One-sided Limits

Lxf

Lxf

cx

cx

=

=

−

+

→

→

)(lim

)(lim Limit from the right

Limit from the left

Find the following limits

1lim

1lim

1lim

1

1

1

−

−

−

→

→

→

−

+

x

x

x

x

x

x

1

0

D.N.E.

D.N.E. RightLeft ≠

Step Functions “Jump”

Greatest Integer [ ]xxf =)(

[ ][ ][ ]=

=

=

→

→

→

+

−

x

x

x

x

x

x

0

0

0

lim

lim

lim -1

0

D.N.E.

RightLeft ≠

32,1

21,52 <<−

≤≤−−

xx

xxg(x)=

=

=

+

−

→

→

)(lim

)(lim

2

2

xg

xg

x

x( )( )=−

=−

+→

→ −

1lim

5lim

2

2

2

x

x

x

x3

3

∴ g(x) is continuous at x = 2

Is g(x) continuous at x = 2?

Intermediate Value Theorem

If f is continuous on [a,b] and k is any numberbetween f(a) and f(b), then there is at least onenumber c in [a,b] such that f(c) = k

[ ]a b

f(a)

f(b)

k

In this case, howmany c’s are therewhere f(c) = k?

3

Show that f(x) = x3 + 2x –1 has a zero on [0,1].

f(0) = 03 + 2(0) – 1 = -1

f(1) = 13 + 2(1) – 1 = 2

Since f(0) < 0 and f(1) > 0, there mustbe a zero (x-intercept) between [0,1].