CONTINUITY
A function is continuous at a point when the
graph passes through the point without a break.
A graph that is discontinuous at a point
has a break of some type at that point.
Can you draw
the graph without
lifting your
pencil???
A. Continuous on the domain.
B. Point Discontinuity (at x = 1)
C. Jump Discontinuity (at x = 1)
D. Infinite Discontinuity (at x = 1)
The function f(x) is continuous at x = a if
f(a) is defined and if lim f ( x ) exists, it must equal f(a)
xa
otherwise, f(x) is discontinuous at x = a.
THIS IS THE THREE PRONG RULE!!
Example 
Test for continuity at each given value of the domain:
y
x
Value
Is f(a) defined?
Does lim f (x ) exist ?
xa
Does lim f (x )  f (a) ?
xa
Continuous at x=a?
x=0
x=1
x =2
x=5
Points to Note:
1. All polynomial functions are continuous.
f (x)
2. A rational function, h( x ) 
, is continuous at x = a if g(a)  0.
g( x )
3. A rational function in simplified form is discontinuous at the zeros
of the denominator.
Example 
Test the continuity of each function at x = 2.
a)
f(x) = x2
c)
f (x) 
x2 4
, x  2; f(2) = 2
x2
b)
f (x) 
x2 4
x2
d)
f (x) 
1
x2
Example 
a)
Example 
Determine all value(s) of x for which each function is continuous:
f (x) 
1
x2  x
b)
f (x) 
1
c)
x2 1
f (x)  x  5
Determine if the following function is continuous on its domain:
– x – 4, if x  –1
g(x) =
2x – 1, if –1 < x < 1
y
4 – x2, if x  1
x
Homework: p.51–53 #1–8, 10–14