BackContinuity and the Intermediate Value Theorem
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Continuity
Definition of Continuity at a Point
A function f(x) is said to be continuous at x = a if all three of the following conditions are satisfied:
The limit exists.
The function is defined at the point: exists.
The value of the function at the point equals the limit: .
If any of these conditions fail, f is discontinuous at x = a, and we say there is a discontinuity at that point.
Continuous Function
A function f(x) is continuous if it is continuous at every point in its domain. For a function defined on a closed interval , it must be right-continuous at and left-continuous at .
Right-continuous at a:
Left-continuous at b:
Example: The function is continuous on .
Types of Discontinuities
Discontinuities in a function can be classified as follows:
Type | Description | Graphical Feature |
|---|---|---|
Removable Discontinuity (Hole) | The limit exists, but is not defined or . | Open circle (hole) in the graph. |
Jump Discontinuity (Gap) | The left and right limits exist but are not equal. | Sudden jump in the graph. |
Infinite Discontinuity (Asymptotic) | The function approaches infinity near . | Vertical asymptote. |
Describing Continuity on Intervals
To determine where a function is continuous, consider the domain restrictions:
Fractions: Denominator
Even Roots: Inside
Logarithms: Inside
Example: For , the function is continuous for all .
Extending Continuity
Sometimes, a function can be made continuous by redefining it at points of discontinuity, often using a piecewise definition.
Example: Extend continuity to by defining as the limit as :
So, define for continuity at .
Piecewise Continuity Examples
To ensure continuity for piecewise functions, set the left and right limits equal at the transition points.
Example 1: Find so that is continuous at .
Set :
Example 2: For what values of and is continuous for all ?
Set the limits equal at and and solve for and .
Intermediate Value Theorem (IVT)
Statement of the IVT
If a function f is continuous on the closed interval and is any number between and , then there exists at least one number in such that .
The IVT is often used to guarantee the existence of zeros (x-intercepts) by showing that a continuous function's values change sign at the endpoints of an interval.
Application of the IVT
Check that the function is continuous on the interval.
Verify that and have opposite signs (for zeros).
Example: Which of the following functions are guaranteed a zero by the IVT on ?
Only is continuous on and changes sign, so the IVT guarantees a zero.
