There are a great many functions that fail the horizontal line test (HLT) which nevertheless have an inverse function.

• fails HLT but is a pretty reasonable inverse for it (just be careful with the negative and positive issue)
• also fails pretty badly, any horizontal line with will hit infinitely many times

How do we resolve this?

Restrictions

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We know that the inverse of sin function exists, it is right on the calculator, which clearly contradicts with my statement earlier. This contradiction can be resolved using the notion of restriction.

Restrictions allow us to eliminate any region in Dom(f) that cause f to fail to be one-to-one: that is, we choose a subset so that is an injection. Now that we have an injective f, we can then find the inverse function f.

Inverse Sine Function

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The graph of a sine function is given,

If one considers the whole function, it obviously fails the HLT. However, if we restrict the domain of the sine function to the closed interval, , it passes the HLT and so we have an invertible function. The inverse of this restricted function is known as the with intervals domain and range as respectively.

Winding Map

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A winding map is a function that goes from to the unit circle in the defined by

The output of this function is unusual in that the output were ordered pairs. Basically, we have an ordered pair, in which the second element is an ordered pair. Another way of expressing a winding map:

Characteristic Functions

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There are other interesting functions we will encounter, all of which are aspects of the characteristic functions of a subset.

Consider a codomain, such that if an input x is in the set S (a subset of the domain D) the function will indicate this by returning 1, otherwise it will return 0. The function which has this behavior is known as , and is called the characteristic function of the subset S. These function is formally defined as:

Iverson bracket notation

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Another interesting function is the Iverson functions, uses a shorthand . It sends an input x that makes its argument true to 1, and any inputs that make false to 0. Formally defined as,

Iversion brackets are particularly useful in expressing and simplifying sums. We can write the number of all even numbers less than 25 as, . The arguments restricts the number i to be divisible by 2. This is just basically counting the number of even numbers less than 25.

For this example: , we are counting the numbers less than 25 that are divisible by 2 and divisible by 3 but not divisible by 6.

Kronecker delta

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Kronecker delta can be considered as a special case of the idea inherent in Iverson bracket. The function is written as that takes two input (i,j) such that if i and j are equal the function is valued 1.

The Iverson bracket equivalent of Kronecker delta is .

_{Disclaimer: this is a summary of section 6.6 from the book A Gentle Introduction to the Art of Mathematics: by Joe Fields, the content apart from rephrasing is identical, most of the equations are screenshots of the book and the same examples are treated.}

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1. A Gentle Introduction to the Art of Mathematics by Joe Field

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