Functions and Continuity
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Functions defined piecewise on their domain
Regarding the reciprocal function[Equation goes here - download the original pdf to see it.] we could define a function that took a value at x = 0. For example [Equation goes here - download the original pdf to see it.] This definition fills in the missing gap when x = 0, though it does so in an arbitrary way and we could have specified that g took any other value when x = 0. Furthermore, the definition of g does not "bridge the gap" created by the singularity at x = 0 for the reciprocal function f. There is no way to join the two halves of f together so that the singularity is removed. When there is a gap like this we say that the function is discontinuous. When there is no "gap" like this, then the function is continuous. Whether a function is continuous or discontinuous is important in mathematics and we need to develop rules for determining this property. The function g has also been constructed from two definitions. Each definition applies to a different part of the domain of g. So [Equation goes here - download the original pdf to see it.] on all values of the domain with the exception of x = 0, and [Equation goes here - download the original pdf to see it.] if x = 0. The function g comes in two pieces and is said to have been defined piecewise on its domain. Intuitively, a function is continuous if its graph consists of one unbroken curve. If you were drawing the graph with a pencil you would draw the graph as one curve or line without lifting your pencil. This is the intuitive notion of a continuous graph. A function can be discontinuous in basically two different ways. Firstly, when a function contains a reciprocal, then it may have a singularity. The singularity means that the graph has asymptotes around the singularity, so it is not possible to join the two halves of the graph together around the singularity. Singularities create discontinuous graphs. Now there is a second way in which continuity can fail. When we define a function piecewise on its domain, we are joining two functions together. In that case, the functions may either join up continuously, or there may be a discontinuity. This is best shown by example. Example (6) The functions g and h are defined piecewise on the domain as follows [Example goes here - download the original pdf to see it.] The sketch for g shows that g remains as an unbroken "curve" - the two halves are joined together continuously at x = 0, and g is continuous throughout its domain. The sketch for h employs the convention that an unfilled circle is used to represent a point that is not included in the graph, and a filled circle represents a point that is included in the graph. When x = 0 the function h takes the value 1 because [Equation goes here - download the original pdf to see it.] , so the point [symbol] is included in the graph and is shown by the filled circle. When [equation] the function h is defined by [Equation goes here - download the original pdf to see it.] and h takes the value 2 at every point when [symbol] except x = 0. So the point [symbol] is not included in the graph and is represented by the unfilled circle. The graph of h shows intuitively that h is discontinuous at x = 0; the graph "jumps" from the value 2 to the value 1 just around x = 0. (We say, "in the neighbourhood of x = 0".) So h is a discontinuous function because the two pieces of h have not been joined together in a way that makes it continuous. However, in this example we are using "intuitive" arguments to prove that one function is continuous and another is discontinuous in the neighbourhood of some point. We need to replace the intuitive argument by something more analytical For this purpose we will need to introduce the idea of a limit. However, when dealing with arguments about limits we employ two types of argument: formal or rigorous arguments and informal arguments. The formal arguments are quite technical, and in this chapter we will in fact use primarily informal arguments about limits to make statements about the continuity of functions. Yet, when we introduce the idea of a limit it is useful to give some idea of what the formal definition of a limit might be. Arguments using formal limits are reserved to another chapter.
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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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