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Section 2.5 Continuity
Continuity describes gradual change both in everyday life and in mathemat- ics. Many properties of continuity seem obvious but they are not always easy to prove. For now we will rely on our geometric intuition, accept the results, and see how they are used as we continue our journey through calculus. In this section, all functions are deÖned either on a single interval or a union of intervals. An interval may be open, closed, or half-open. (See Appendix A in Stewart if you need a review of intervals.) A function f (x) is continuous at a if
lim x!a f (x) = f (a) :
In order for this limit statement to make sense it is understood that
- a must be in the domain of f:
- f must have a (two-sided) limit at a: The related notions of continuous from the right ( resp., left) at a are deÖned by the one-sided limit relations
lim x!a+ f (x) = f (a) and lim x!a f (x) = f (a) ;
respectively. Apparently a function f (x) is continuous at a if and only if it is both con- tinuous from the right and continuous from the left at a: A function is continuous (on its domain) if it is continuous at each point of its domain, with one-sided continuity at endpoints of intervals in its domain. Since
limx!a p (x) = p (a) for any polynomial p (x) ; and limx!a r (x) = r (a) for any rational function r (x) and any a in its domain
we see that
All polynomials and all rational functions are continuous.
Likewise, since n
p x! n
p a as x! a for any a in the domain of n
p x;
All n-th root functions are continuous.
A function f is discontinuous at a if it is not continuous at a. The simplest type of discontinuity occurs when the one-sided limits exist but are not equal at a. Then there is a jump or gap in the graph at a; and f is said to have a jump discontinuity at a: Algebraic Properties of Continuous Functions The algebraic limit laws and the deÖnition of continuity immediately give corresponding useful laws of continuity.
- The sum of two continuous functions is a continuous function.
- The di§erence of two continuous functions is a continuous function.
- A constant times a continuous function is the continuous function.
- The product of two continuous functions is a continuous function.
- The quotient of two continuous functions is a continuous function (on its domain).
These algebraic laws of continuity apply to Önite sums, di§erences, prod- ucts, and quotients of continuous functions. They also are valid with continuity replaced by continuity at a point and with continuity replaced by one-sided continuity. Composite Functions Complicated functions are frequently built up from simpler functions by substitution of one function into another. For example, substitution of the polynomial 1 x^2 into the square root function
p x yields the more complicated function
p 1 x^2. If f (x) and g (x) are functions, we deÖne a new function, denoted by f g and called the the composition of f and g, by
(f g) (x) = f (g (x))
for all x for which the right side makes sense; that is, for all x such that x is in the domain of g and g (x) is in the domain of f: For example, if f (x) =
p x and g (x) = 1 x^2 ; then
(f g) (x) = f (g (x)) = f
1 x^2
p 1 x^2 :
What is the domain of f g in this case? In this example, both f and g are continuous functions and it should be reasonably clear that their composition f g also is continuous. This illustrates a general fact:
The composition of continuous functions is continuous.
The Intermediate Value Theorem The graph of a continuous function deÖned on an interval I has no ìbreaksî or ìgapsî. A precise mathematical version of this fundamental property of continuity is the
Intermediate Value Theorem (IVT): Let f be a continuous function on an in- terval I and let a and b be any two points in I: Let r be any number between f (a) and f (b) : Then there is a number x between a and b such that f (x) = r:
Informally stated, a continuous function does not skip over any values.
A Substitution Limit Law
