Continuity

Section 4.3 Continuity

Subsection 4.3.1

Figure 4.3.1. Continuity

A function is continuous at a point \(a\) in its domain if

\begin{equation*} f(a) = \lim_{x \rightarrow a} f(x)\text{.} \end{equation*}

I have already mentions these kinds of limit: in the calculation of limits, the first step was to try to evaluate the function at the limit point. In this step, I am implicitly relying on the continuity of the standard functions. Happily, I was justified in doing so: all the standard elementary functions (polynomials, rational function, algebraic functions, trig, exponentials) are continuous on their domains. The phrase ‘on their domains’ is important; where there is a break in the domain, the function cannot be continuous. Continuity only happens inside the domain of a function. Visually, the graph of a continuous function is connected; it can be drawn without lifting your pen (pencil, chalk, etc).

Subsection 4.3.2 Intermediate Value Theorem

Though it has a formal name, the Intermediate Value Theorem is a very sensible and obvious result of continuity. Formally stated, the theorem says that if \(f\) is a continuous function on an interval \([a,b]\) and \(\alpha\) is a real number with \(f(a) \lt \alpha \lt f(b)\) or \(f(a) > \alpha > f(b)\text{,}\) then there exists a number \(c \in (a,b)\) such that \(f(c) = \alpha\text{.}\)

Rephrased, the theorem says that a continuous function cannot skip any values. If \(f(a) = 0\) and \(f(b) = 1\text{,}\) then the function must output all values between \(0\) and \(1\) as the input goes from \(a\) to \(b\text{.}\) Continuity means the graph is connected; it can’t jump over any intermediate values. While this seems like a very obvious result, it is quite useful to have it formally stated. Many other mathematical proofs make use of the Intermediate Value Theroem for continuous functions.

Figure 4.3.2. Intermediate Value Theorem

Subsection 4.3.3 Piecewise Functions

Figure 4.3.3. The Heaviside Function

All of the familiar elementary functions are continuous on their domains. One might ask: why worry about continuity at all? A common place where continuity becomes an issue is in piecewise functions. These are functions which have different expressions or definitions over different pieces of their domain.

Piecewise functions have special notation for their definitions. Consider a function on \(\RR\) which has two definitions, one for \(x \lt a\) and another \(x \geq a\text{.}\) Then the function is written

\begin{equation*} f(x) = \begin{cases} g(x) \amp x \lt a \\ h(x) \amp x \geq a \end{cases} \text{.} \end{equation*}

With this notation, I can easily indicate how the function behaviours on various pieces of its domain.

A very well-known example is the Heaviside function.

\begin{equation*} h(x) = \begin{cases} 0 \amp x \lt 0 \\ 1 \amp x \geq 0 \end{cases} \end{equation*}

The Heaviside function is very frequently used to model switches: it suddenly changes from off (0) to on (1) at \(x=0\text{.}\) This is a discontinuous change, as the function jumps from \(0\) to \(1\) without going through any intermediate value.

In the definition of a piecewise function, there can be more than two pieces and the conditions on \(x\) can be more complicated than inequalities. Here is a piecewise function with three pieces defined on intervals.

\begin{equation*} f(x) = \begin{cases} x^2 -1 \amp x \in (-5,0) \\ x^2 + 1 \amp x \in [0,3] \\ 3x-5 \amp x \in (3,7) \end{cases} \end{equation*}

The domain of this function is \((-5,7)\text{,}\) which is the union of all the intervals of definition in the piecewise expression. It’s graph is Figure 4.3.4.

Piecewise functions can be extremely strange.

\begin{equation*} f(x) = \begin{cases} 1 \amp x \in \QQ \\ 0 \amp x \notin \QQ \end{cases} \end{equation*}

This function depends on whether its input is rational or irrational, returning one and zero respectively. It is a horendously discontinuous functions, with ones and zeros everywhere and no intermediate values whatsoever. However, it is still a useful function, in that it, somehow, picks our all the rational numbers by assign to them the value \(1\text{.}\)

Figure 4.3.4. A Discontinuous Piecewise Function

Subsection 4.3.4 Continuity of Piecewise Functions

It is important to be able to check if a piecewise function is continuous at its crossover point. Consider a piecewise function with a break at \(x=a\text{.}\)

\begin{equation*} f(x) = \begin{cases} g(x) \amp x \lt a \\ h(x) \amp x \geq a \end{cases} \end{equation*}

It is natural to ask if \(f\) is continuous at \(x=a\text{.}\) In order to investigate this, I need to look at the function value and the limits from both sides. The function is continuous if all three are the same.

\begin{equation*} \lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = f(a) \end{equation*}

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