1. C4 - Discontinuities
This unit discusses three types of discontinuities (breaks in a graph). Each one has
specific algebraic characteristics that result in a certain type of behaviour on a graph.
Infinite Discontinuities
This type of discontinuity occurs at a vertical asymptote. They are located by finding
values of x that yield a divide by zero error. In other words, a function will yield a result
of where a 0.
This graph of has
vertical asymptotes at 2 and -2.
The infinite discontinuity exists at 2
and -2.
Algebraic Example 1
Determine the infinite discontinuity in the function .
2. Algebraic Example 2
Determine the infinite discontinuity in the function .
Jump Discontinuities
This type of discontinuity is typically a step function, where the graph "jumps" from one
step to another.. They are located by finding values of x wher the absolute value of a
function changes from positive to negative.
This graph of becomes
The jump discontinuity occurs where
the function steps up from -1 to 1, at 3
Algebraic Example
.
3. Removable Discontinuities
This type of discontinuity often occurs at a "hole" in the graph, or the point where the
graph is indeterminate. They are located by finding values where both the numerator and
denominator equal zero. In other words, a function will yield a result of when f(x) =
the removable discontinuity.
Graphing Example
The graph of is shown,
yet the graph looks like the
graph of .
The description of the graph can be
shown as follows:
The graph has a removable
discontinuity at 2
Algebraic Example 1
Determine where the removable discontinuity exists in .
The removable discontinuity exists at x = -3.
Summary Example 2
Determine all discontinuities that exist in .
