Functions and Continuity
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Quotients
When one function is divided by another, then we have the possibility of singularities. A singularity may occurs whenever a value of x would lead to a zero in the denominator of the quotient. Let [Equation goes here - download the original pdf to see it.] Suppose that when x = a the function [equation] and suppose also that at this point [Equation goes here - download the original pdf to see it.]. Then at this point the quotient cannot be defined and [Equation goes here - download the original pdf to see it.] cannot take a value. Strictly speaking the point x = a cannot belong to the domain of [Equation goes here - download the original pdf to see it.]and we should specify that [Equation goes here - download the original pdf to see it.] is defined on a domain that does not include x = a. Alternatively, we could add a value for [symbol] by defining it piecewise [Equation goes here - download the original pdf to see it.] In practice we speak of [Equation goes here - download the original pdf to see it.] being defined on the whole domain (say ) and assume that it is clear from context that [Equation goes here - download the original pdf to see it.] is undefined at any value where [Equation goes here - download the original pdf to see it.]. Example (8) Name any points at which the following functions are discontinuous [Example goes here - download the original pdf to see it.] As the (a) above indicates there exists the possibility that a quotient may have a limit even at a point where the denominator takes a zero value; that is, if the numerator simultaneously is zero at this point. However, this alone would not be enough to show that a quotient was continuous, because the function has to converge from above and below on zero, and not just equal zero at that point. Nonetheless, here is an important example of when a quotient has a limit and is continuous at a given point, even though the denominator is zero at that point. Example (9) Prove that the function [Equation goes here - download the original pdf to see it.] is continuous at x = 0. Furthermore, show that the limit [Example goes here - download the original pdf to see it.]
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Contents of
Functions and Continuity
1 Functions
2 Graph
3 Inverse of a Function: monotone increasing or decreasing functions
4 The reciprocal function - singularities
5 Functions defined piecewise on their domain
6 Limits
7 Formal definition of a limit
8 Formal definition of the limit of a function at a point x = a
9 Informal arguments
10 Continuity
11 Combining limits
12 Quotients
13 Image set
14 Inverse image
15 Odd and even functions
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