Page 5 - mathematics

Types of Discontinuity 2- Jump discontinuity

1- Removable discontinuity y lim f x  L1, lim f  x  L2

a xa xa
y
lim f x Exist f(a) is not defined x L1  L2
x
xa or y

f(a) is defined to
be a number other
than the value of
limit.

x
a

3- Infinite discontinuity Example

Discuss the continuity of f(x) at x = 2, where

lim f x    or lim f x    or both x2  x  2, x2
x a  x a 
f x   x2

 1, x2

y Solution

f 2 1

lim f x  lim x2  x  2  0  lim  x  2 x  1  3
x2 x  2 0
x x2 x2 x2

lim f  x   f 2 Removable discontinuity

x 2

Example Example

Discuss the continuity of f(x) at x = 0, where

f  x  1, x0
0, x0

Solution Solution

lim f  x 1  lim f  x  0 X=-3 Removable discontinuity
x0 x0 X=0 Infinite discontinuity
X=5 Jump discontinuity
Jump discontinuity X=2 The function is continuous

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