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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].