Embed Size (px)
Citation preview
Continuity
(Section 2.6)
When all of the answers are YES, i.e.
,
we say f is continuous at a.
)()(lim afxfax
Continuitylimit matches function value
1. Is the function value f(a) defined?
2. Does the limit as xa exist?
3. Does the limit match the function value?
Continuity checklist
A continuous function:
Graph has no holes, breaks, or jumps. You could draw it without lifting your pencil.
When the x-values are close enough to each other, so are the corresponding function values.
Continuity
2)(lim1
xfx
Limit as x1 exists and matches function value, so function is continuous at x=1.
4321)( xxxxf
limit matches function value
2)(lim1
xf
x
2)(lim1
xfx
2)1( f
Jump discontinuity(at x=2)
2 if6
2 if)(
xx
xxxf
Discontinuitylimit does not match function value
2)(lim2
xf
x
4)(lim2
xfx
4)2( f
)(lim2
xfx
Does Not Exist
Infinite discontinuity(at x=3)
2)3(
1)(
x
xf
Discontinuitylimit does not match function value
)(lim3
xfx
)(lim3
xfx
)(lim3
xfx
)3(f undefined
Removable discontinuity(at x=5)
)5(
)5)(2()(
x
xxxf
Discontinuitylimit does not match function value
3)(lim5
xfx
3)(lim5
xf
x
3)(lim5
xfx
)5(f undefined
The discontinuity can be removed by defining f(5) to be 3.
Removable discontinuity(at x=0)
x
xxf
sin)(
Discontinuitylimit does not match function value
1)(lim0
xfx
1)(lim0
xf
x
1)(lim0
xfx
)0(f undefined
The discontinuity can be removed by defining f(0) to be 1.
Removable discontinuity (at x=1)
1 if1
1 if4)(
2
x
xxxf
Discontinuitylimit does not match function value
3)(lim1
xfx
3)(lim1
xf
x
3)(lim1
xfx
1)1( f
Limit does not match function value
The discontinuity can be removed by redefining f(1) to be 3.
Oscillating discontinuity(at x=0)
xxf
1sin)(
Discontinuitylimit does not match function value
)(lim0
xfx
Does Not Exist
)(lim0
xfx
Does Not Exist
)(lim0
xfx
Does Not Exist
)0(f undefined
There is no way to “repair” the discontinuity at x=0.
Continuitylimit matches function value
1. Is the function value f(a) defined?
2. Does the limit as xa exist?
3. Does the limit match the function value?
Continuity checklist
)()(lim afxfax
If all of the answers are YES, i.e.
,
then f is continuous at a.
A continuous function:
Graph has no holes, breaks, or jumps. You could draw it without lifting your pencil.
When the x-values are close enough to each other, so are the corresponding function values.
29)( xxf continuous at x = 1
left-continuous at x = 3
right-continuous at x = -3
0)3(
0)(lim3
f
xfx
0)3(
0)(lim3
f
xfx
228)1(
228)(lim1
f
xfx
922 yx
Circle with radius 3,centered at origin:
Solve for y:
29 xy
Top half of circle:29 xy
Bottom half of circle:29 xy
Review: Circles
Left and Right Continuityleft-hand or right-hand limit matches function value
