Continuity
1
Definition of Continuity
A function f (x) is said to be continuous at a point x = a if and only if lim f (x) and f (a) both exist and
x→a
lim f (x) = f (a)
x→a
f (x) is said to be left continuous (or “continuous from the left”) at x = a if and only if lim f (x) and f (a)
x→a−
both exist and
lim f (x) = f (a)
x→a−
f (x) is said to be right continuous (or “continuous from the right”) at x = a if and only if lim f (x) and
x→a+
f (a) both exist and
lim f (x) = f (a)
x→a+
A function f (x) is said to be continuous on an open interval I if and only if f is continuous at every point in
I. We include the left endpoint of I if f is right continuous there, and we include the right endpoint
of I if f is left continuous there (we always want to approach the endpoints of an interval through x-values
in the interval).
1.1
Discontinuities
There are two types of discontinuities:
1. Removable. A discontinuity is removable if we can [re]define the function at a single value of x and
make it continuous there. Graphically, a removable discontinuity will look like a hole in the graph.
2. Non-removable. A discontinuity is non-removable if we cannot [re]define the function at a single value
of x and make it continuous there. Graphically, a non-removable discontinuity will look like either
• a vertical asymptote, or
• a jump in the graph.
1.2
Properties
1. Suppose f and g are continuous functions on some interval I, and c is a constant. Then the following
are also continuous:
• f ±g
1
• c·f
• f ·g
f
(if g 6= 0)
•
g
2. If g is continuous at a and f is continuous at g(a), then f ◦ g is continuous at a (i.e., “the composition
of continuous functions is continuous”).
3. The following types of functions are continuous at every point in their domains: Polynomials,
rational, radical, exponential, logarithmic, trigonometric.
1.3
Examples
y
6
4
2
-6
-4
-2
2
4
6
x
-2
-4
2
The Intermediate Value Theorem (IVT)
“Let f (x) be a continuous function on a closed interval [a, b] (such that f (a) 6= f (b)), and let N be any number
between f (a) and f (b). Then there exists a number c in the open interval (a, b) such that f (c) = N .”
We can use the IVT to verify or find the approximate location of roots/zeros of a continuous function, or to
approximate solutions to an equation involving continuous functions.
2