One of the items in our official analysis list is continuity.
Graphically, continuity looks like no breaks in the graph.
Algebraically, it is not so easy to describe.
The algebraic description will need to dsecribe somehting our eyes see very easily, but
not so easy to put into words, or algebra.
Luckily, all of our elementary functions are continuous functions.
- Power Functions
-
Polynomials
- Constant Functions
- Linear Functions
- Quadratic Functions
- Rational Functions
- Radical Functions
- Exponential Functions
- Shifted Exponential Functions
- Logarithmic Functions
- Sine Functions
- Cosine Functions
- Tangent Functions
- Absolute Value Functions
They are all continuous functions.
Operations
Creating functions through our normal operations, maintains continuity.
- a Sum;
- a Difference;
- a Product; or
- a Quotient;
They are all continuous functions.
Composition
Creating functions through composition, maintains continuity.
Compositions of continuous functions are continuous functions.
Discontinuities
If all of our elementary functions are continuous, and all of their combinations and
compositions are continuous, then how are we going to study discontinuities?
The only way to study discontinuities is to create them with piecewise defined
functions.
So, that’s what we’ll do.
And, that is why piecewise defined functions are important to us.
They are our only tool for studyig discontinuities.
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more examples can be found by following this link
More Examples of Analysis