Step By Step Calculus » 9.4 - Continuous Functions and the Intermediate Value Theorem

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Synopsis
To say a function is continuous intuitively tells us that its graph can be drawn without lifting the pencil off the page. There are exceptions to this, as we show in the detailed discussion, but it is a good intuitive start to understanding the concept. If a function is continuous at a point x = ax = a, then we can bring limits at aa inside the function:
\displaystyle \lim_{x\to a} f(x)=f\left(\lim_{x\to a} x\right)=f(a) \; .
\displaystyle \lim_{x\to a} f(x)=f\left(\lim_{x\to a} x\right)=f(a) \; .
Based on our recent discussion of limits we can formally define three different types of continuity:
  • A function f: D_f\rightarrow E_ff: D_f\rightarrow E_f is continuous from the right at a\in D_fa\in D_f if \lim\limits_{x\rightarrow a^+}f(x)=f(a)\lim\limits_{x\rightarrow a^+}f(x)=f(a).
  • A function f: D_f\rightarrow E_ff: D_f\rightarrow E_f is continuous from the left at a\in D_fa\in D_f if \lim\limits_{x\rightarrow a^-}f(x)=f(a)\lim\limits_{x\rightarrow a^-}f(x)=f(a).
  • A function is continuous at a\in D_fa\in D_f if it is continuous from both the left and the right.
In particular a function is continuous at x = ax = a if and only if the function satisfies three requirements:
  • f(a)f(a) must be defined.
  • \lim\limits_{x\rightarrow a^{+}}f(x)\lim\limits_{x\rightarrow a^{+}}f(x) must exist if you can approach aa from above from within D_fD_f. In this case, we must have \lim\limits_{x\rightarrow a^{+}}f(x)=f(a)\lim\limits_{x\rightarrow a^{+}}f(x)=f(a).
  • \lim\limits_{x\rightarrow a^{-}}f(x)\lim\limits_{x\rightarrow a^{-}}f(x) must exist if you can approach aa from below from within D_fD_f. In this case, we must have \lim\limits_{x\rightarrow a^{-}}f(x)=f(a)\lim\limits_{x\rightarrow a^{-}}f(x)=f(a).
Continuity may also be defined on an interval, or on subsets of D_fD_f. Specifically, “ff is continuous on its domain D_{f}D_{f} if it is continuous at each a\in D_fa\in D_f. If A\subseteq D_f,A\subseteq D_f, then we say ff is continuous on AA if the restriction of ff to A A is continuous on AA.”
There are continuity rules: Given that c\in {\mathbb R}c\in {\mathbb R} is a constant, and that ff, gg are continuous at aa from one side or both, then the following functions are also continuous in the same sense:
Continuity Linearity Law:cf+gcf+g
Continuity Quotient Law:\frac{f}{g}\frac{f}{g} provided that g(a)\neq 0g(a)\neq 0
Continuity Product Law:fgfg (note that this is the product, not the composition f\circ gf\circ g)
Continuity Power Law:f^pf^p for p \in \mathbb{R}, p \in \mathbb{R}, if p>0 p>0 and f(x)\geq 0f(x)\geq 0 in an appropriate neighborhood of aa.
We can use these laws to show that polynomials, rational functions and ratios of functions involving roots are continuous on their domains. Also, we will show that trigonometric functions, the exponential function, the natural logarithm (in fact any logarithm) and hyperbolic functions are all continuous on their respective domains. Notice that with ratios of functions, the domain would exclude vertical asymptotes, and these are certainly points where the function is not continuous. In addition, with rational functions we have to be careful of expressions like \dfrac{x^{2} -4}{x-2} \dfrac{x^{2} -4}{x-2} which is equal to x + 2x + 2 everywhere but at x =2x =2 where it is undefined. Therefore we must be aware that its domain is not the entire real line.
The continuity of the composition of two continuous functions is determined by the following rule:
Composite Rule:f\circ gf\circ g is continuous at aa if (i) gg is continuous at aa and (ii) ff is continuous at g(a)g(a).
Continuity of the inverse of a continuous function is determined by the following rule:
Inverse Function Rule: If the function ff is continuous and invertible on an interval II, then for any a \in I, \quad f^{-1} a \in I, \quad f^{-1} is continuous at f(a) f(a).
The Intermediate Value Theorem is a simple and powerful tool for approximations involving continuous functions. It states that if ff is continuous on [a,b][a,b] and kk is a value strictly between f(a)f(a) and f(b)f(b) then there is at least one number c\in \left(a,b\right)c\in \left(a,b\right) such that f(c)=kf(c)=k. Bolzano’s bisection method can be used to approximate the value of kk to any prescribed degree of accuracy. In particular, if f(a)f(a) and f(b)f(b) have different signs then there is at least one root of ff (a point where f(x) = 0f(x) = 0 ) between aa and bb.