WOLFRAM|DEMONSTRATIONS PROJECT

Discontinuity

​
range
20
F(x)(x-a)(x-b)
a
-2
b
-1
divide by x-c?
F(x)
(x-a)(x-b)
x-c
c
2
infinite or removable discontinuity at c
jump at d?
G(x)
F(x)
x<d
e+F(x)
x≥d
d
-1
e
5
jump discontinuity at d
remove r?
r
1
removable discontinuity at r
vertical asymptote
For a value
f(s)
, let
-
L
=
lim
x
-
s
f(x)
(the limit from the left) and
+
L
=
lim
x
+
s
f(x)
(the limit from the right).
If
-
L
=
+
L
=
f(s)
, the function is continuous at
x=s
.
If
-
L
=
+
L
≠
f(s)
, the function has a removable discontinuity at
x=s
.
If
-
L
≠
+
L
, and both values are finite, the function has a jump discontinuity at
x=s
.
If
-
L
≠
+
L
, and one or both values is infinite, the function has an infinite discontinuity at
x=s
. This is also called an essential discontinuity.