Figure 10-4: Continuity condition
Suppose we have a function that maps real numbers to real numbers. On a graph the function is continuous if it is a single unbroken curve with no gaps. A continuous function is defined to be one in which arbitrarily small changes in the independent variable produce arbitrarily small changes in the value of the function. If small changes in the independent variable produce a jump in the value of the function or cause the value of the function to be undefined, the function is discontinuous. Rational functions and functions defined piecewise are likely to have points of discontinuity.
Rational functions may be discontinuous at the points where the function is undefined; division by zero is the usual cause. Consider rational function:
x + 2
f(x) = ————————
x2 - 4
The independent variable x is in the denominator and there is one or more values of
x that will make the function undefined. For f(x), the domain is all real
numbers except -2 and +2. At these points the function is discontinuous.
Piecewise functions may be discontinuous at the points where the function definition changes. A function or curve is piecewise continuous if it is continuous on all but a finite number of points at which certain matching conditions are sometimes required.
Consider the function f: R -> R. Let c be in the domain of f. Then f is continuous at the point c if and only if there is some number δ > 0 such that for all x in the domain with:
the value of f(x) will satisfy:|x - c| < δ
for any number ε > 0 however small. That is, for any arbitrary choice of ε > 0, a function is continuous if there is a number δ > 0 such that a point on the graph will be within the rectangle shown in the figure below.|f(x) - f(c)| < ε
Figure 10-4: Continuity condition
The following functions are continuous in the set of real numbers:
f(x) = ax2 + bx + c f(x) = bx f(x) = log x for x > 0 f(x) = √x for x ≥ 0 f(x) = |x| f(x) = 1/(x2 + 1)
The following functions are discontinuous at one or more points in the set of real numbers:
f(x) = 1/x discontinuous at x = 0
f(x) = tan(x) discontinuous at x = (2n-1)π/2, n is an integer
The following functions are discontinuous at the point between each piece of the function:
Unit step function
0 if x < 0
u(x) = 0.5 if x = 0
1 if x > 0
Floor function
Ceiling functionfloor(x) = greatest integer ≤ x
Round functionceil(x) = smallest integer ≥ x
round(x) = integer nearest to x
Even integer if fractional part = 0.5
Rounding is ambiguous for half-integers; by convention half-integers are rounded
to even numbers in order to avoid statistical biasing.
If two functions f(x) and g(x) are continuous, then the following are continuous:
f(x) + g(x) f(x) - g(x) f(x)·g(x) f(x)/g(x) if g(x) ≠ 0 for all x in the domain
If the function f(x) is continuous on a closed interval [a, b] then for every real number d between f(a) and f(b), there is at least one number c in the open interval (a, b) such that f(c) = d.
If the function f(x) is continuous on a closed interval [a, b] whose values have different signs at the endpoints of the interval, the graph of f(x) crosses the x-axis at least once in the open interval (a, b). That is, at some point c, the function f(c) must equal zero.
If the function f(x) is continuous on a closed interval [a, b], then their are two numbers, c and d, in this interval such that f(c) ≤ f(x) ≤ f(d) for all x in the interval. That is, f(x) has both a maximum and a minimum in the closed interval [a, b].