WOLFRAM|DEMONSTRATIONS PROJECT

Discontinuity

​
range
F(x)(x-a)(x-b)
a
b
divide by x-c?
F(x)
(x-a)(x-b)
x-c
c
infinite or removable discontinuity at c
jump at d?
G(x)

F(x)
x<d
e+F(x)
x≥d

d
e
jump discontinuity at d
remove r?
r
removable discontinuity at r
vertical asymptote
-10
-5
5
10
-20
-10
10
20

(x+1)(x+2)
x-2
x<-1
(x+1)(x+2)
x-2
+5
x≥-1
when x ≠ 1 or 2

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.