NUMBER SYSTEMS AND ANALYSIS

Real Variables

Contents

Continuity
Algebraic Properties of Continuous Functions
Continuous Functions in [a, b]
Simple Approximations to Continuous Functions
Derivatives
Elementary Special Functions
Differentials
The Chain Rule
Implicit Functions
Integration
The Fundamental Theorem of Calculus
More Properties of Integrals

Continuity

A many-one relation between real numbers is called a real-valued function of a real variable. We write: y = f(x).

A function f is continuous at u when, for all sequences xk which have the limit u, the sequence f(xk) has the limit f(u).

An equivalent definition of continuity is as follows: A function f is continuous at u when, for all epsilon > 0, there exists delta > 0 such that | x - u | < delta implies | f(x) - f(u) | < epsilon.

EXAMPLES

  1. f(x) = x2 is continuous everywhere.
  2. f(x) = | x | is continuous everywhere.
  3. f(x) = 1/x is continuous everywhere except at x = 0. At x = 0 the function has an infinite discontinuity.
  4. f(x) = 1/(x2 - 1) is continuous everywhere except at x = ±1. At x = -1 and x = +1 the function has infinite discontinuities.
  5. f(x) = x / | x | is continuous everywhere except at x = 0. At x = 0 the function has a finite jump discontinuity from f(x) = -1 where x < 0 to f(x) = +1 where x > 0.

Algebraic Properties of Continuous Functions

If f(x) and g(x) are both continuous at u, then:
  1. f(x) + g(x) is continuous at u.
  2. f(x).g(x) is continuous at u.

Continuous Functions in [a, b]

The closed bounded interval [a, b] is the set of all real numbers x such that a < x < b. Important properties of continuous functions in closed bounded intervals are given in the following theorems. The open interval (a, b) is the set of all real numbers x such that a < x < b. Open intervals are also important in real variable analysis.

THEOREM. If f(x) is continuous at every point in [a, b], then f(x) is bounded in [a, b].

PROOF. Suppose f(x) is unbounded in [a, b]. Then for all k there exists xk in [a, b] such that | f(xk) | > k. By the properties of bounded sequences, the sequence x1, x2, x3, ... has a subsequence x1', x2', x3', ... which has a limit x' in [a, b]. Since the sequence | f(x1') |, | f(x2') |, | f(x3') |, ... is unbounded, it cannot have a finite limit at x'. Therefore there exists x' in [a, b] where f(x) is not continuous.

THEOREM. A function which is continuous at every point in [a, b] has a maximum in [a, b].

PROOF. By the previous theorem, if f(x) is continuous at every point in [a, b], then f(x) is bounded in [a, b]. Let M = sup f(x) in [a, b]. Then for all k there exists xk in [a, b] such that M - f(xk) < 1/k. Therefore lim f(xk) = M. The sequence x1, x2, x3, ... has a subsequence x1', x2', x3', ... which has a limit x' in [a, b]. Then lim f(xk') = M. Since f(x) is continuous, we also have lim f(xk') = f(x'). Therefore f(x') = M. In other words f(x) has a maximum at x' in [a, b].

THEOREM. A function which is continuous at every point in [a, b] has a minimum in [a, b].

The proof is similar to the proof of the above theorem

The Intermediate Value Theorem. If f(x) is continuous in [a, b], and if y' is any number between f(a) and f(b), then there exists x' in (a, b) such that f(x') = y'.

PROOF. Let E be the set of points in [a, b] where f(x) < y', and let x' = sup E. Then there is a sequence u1, u2, ... of points in E which has the limit x'. Since f(x) is continuous, we have lim f(uk) = f(x'), where f(x') < y'. Let F be the set of points in [a, b] where f(x) > y'. Then there is a sequence v1, v2, ... of points in F which has the limit x'. Since f(x) is continuous, we have lim f(vk) = f(x'), where f(x') > y'. Therefore f(x') = y'.

Simple Approximations to Continuous Functions

Suppose f(x) is continuous in [a, b]. Then f(x) is uniformly continuous in [a, b]. In other words, for all epsilon > 0 there exists delta > 0 such that | x1 - x2 | < delta implies | f(x1) - f(x2) | < epsilon in [a, b]. Choose epsilon to give the accuracy you want. Make a partition of [a, b] by points x0 = a < x1 < x2 < ... xn-1 < xn = b such that for all k you have xk - xk-1 < delta. Then you can make the following simple approximations to f(x) with accuracy epsilon.

The step function

This function (shown in Fig. 1) is useful for approximate numerical integration. It is defined as follows: g(x) = f [(½).(xk-1 + xk)], where x is in (xk-1, xk).

Fig. 1.

Fig. 1. The step function approximation.

The piecewise linear function

This function (shown in Fig. 2) is useful for approximate numerical differentiation. It is defined as follows: h(x) = f(xk-1) + mk.(x - xk-1 ), where mk = [f(xk) - f(xk-1)] / (xk - xk-1) and x is in [xk-1, xk].

Fig. 2.

Fig. 2. The piecewise linear function approximation.

Derivatives

Let f(x) be a given function. Suppose that, for every sequence xk such that lim xk = x,
f '(x) = lim[f(x) - f(xk)] / [x - xk].
Then the function f '(x) is called the (first) derivative of f(x).

The second derivative of f(x) is the derivative of the first derivative:

f "(x) = lim[f '(x) - f '(xk)] / [x - xk].
The rules of algebra for calculating derivatives are as follows.

If f(x) = u(x) + v(x), then f '(x) = u'(x) + v'(x).

If f(x) = u(x) - v(x), then f '(x) = u'(x) - v'(x).

If f(x) = u(x).v(x), then f '(x) = u'(x).v(x) + u(x).v'(x).

If f(x) = u(x)/v(x), then f '(x) = [u'(x).v(x) - u(x).v'(x)] / [v(x)]2.

Using these rules we can differentiate polynomials as follows:

If p(x) = a0.xn + a1.xn-1 + ... + an-1.x + an, then p'(x) = n.a0.xn-1 + (n - 1).a1.xn-2 + ... + 2.an-2.x + an-1.

Using these rules we can also differentiate rational functions f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.

Elementary Special Functions

The elementary special functions are continuous differentiable functions defined by the following properties:
The exponential:
exp(0) = 1, exp'(x) = exp(x).
The logarithm:
The inverse of the exponential: exp(log(x)) = x, where x > 0, log(1) = 0, log'(x) = 1/x.
The sine:
sin(0) = 0, sin'(0) = 1, sin"(x) = -sin(x).
The cosine:
cos(0) = 1, cos'(0) = 0, cos"(x) = -cos(x).

Differentials

In the equation
y = f(x),
x is the independent variable, f is the function, and y is the dependent variable. Let dx be an independent parameter, which is called the differential of x. Then dy, which is called the differential of y, is defined by
dy = f '(x).dx.
Note that dy depends on two independent quantities: the variable x and the parameter dx.

Suppose that f is a function of n independent variables x1, ... , xn:

y = f(x1, ... , xn).
Let dx1, ... , dxn be the n independent differentials of x1, ... , xn. Then the differential of y is defined by
dy = f1(x1, ... , xn).dx1 + ... + fn(x1, ... , xn).dxn,
where f1, ... , fn are the partial derivatives of  f  with respect to x1, ... , xn. Note that dy depends on 2n independent quantities: the n variables x1, ... , xn and the n parameters dx1, ... , dxn.

Let y = f(x). Then the second differential of y is a new dependent variable defined by

d2y = d(dy) = [f '(x).dx]'.dx = [(f '(x))'.dx + (f '(x)).(dx)'].dx.
Since (f '(x))' = f "(x), and (dx)' = 0, because dx is a parameter independent of x, we have
d2y = f "(x).(dx)2.

The Chain Rule

Suppose y = f(x) and x = g(u). Then y = f[g(u)]. Differentiating we have
dy = f '(x).dx = f '[g(u)].d[g(u)] = f '[g(u)].g'(u).du.
This formula for dy in terms of u and du (instead of x and dx) is called the chain rule.

Implicit Functions

Suppose the relation between n variables is given by
f(x1, ... xn) = 0.
Note that we do not distinguish between dependent and independent variables; all the variables have the same status. Differentiating, we get
f1(x1, ... , xn).dx1 + ... + fn(x1, ... , xn).dxn = 0.
The rate of change of xi with respect to xj can be calculated from this equation by making all the differentials zero except dxi and dxj. The rate of change is then given by:
dxi/dxj = - fj(x1, ... , xn) / fi(x1, ... , xn).

Integration

Divide the interval [a, b] into k subintervals of length hk = (b - a)/k, and let x1, ... , xn be the mid points of the subintervals. Let f(x) be a piecewise continuous function defined in [a, b]. Then the Riemann integral of f(x) from x = a to x = b is
Integral(a, b): f(x).dx = lim [f(x1) + ... + f(xk)].hk, where k = 1, 2, ...
Note: The differential dx is not used in the calculation of the integral; it only shows that x is the variable of integration.

The integral has the following elementary properties:

Integral(b, a): f(x).dx = -Integral(a, b): f(x).dx.

Integral(a, b): [f(x) + g(x)].dx = Integral(a, b): f(x).dx + Integral(a, b): g(x).dx.

Integral(a, b): c.f(x).dx = c.Integral(a, b): f(x).dx, where c is a constant.

If f(x) is even, in other words f(-x) = f(x), then Integral(-a, a): f(x).dx = 2.Integral(0, a): f(x).dx.

If f(x) is odd, in other words f(-x) = -f(x), then Integral(-a, +a): f(x).dx = 0.

The Fundamental Theorem of Calculus

Let F(x) be a function such that F'(x) = f(x). Then
Integral(a, b): f(x).dx = F(b) - F(a).
Simplified Proof. We shall use the same notation as in the definition of the Riemann integral. Let [a, b] be divided into k equal subintervals. Then b = a + k.hk, where hk is the length of each subinterval. Let xk be the mid point of each subinterval.

First, let F(a) be given, and let F(a + hk) = F(a) + f(x1).hk. Then f(x1) = [F(a + hk) - F(a)] / hk = [F(x1 + (½).hk) - F(x1 - (½).hk)] / hk.

Next, for j = 2, ... k - 1, let F(a + j.hk) = F(a + (j - 1).hk) + f(xj).hk. Then f(xj) = [F(a + j.hk) - F(a + (j - 1).hk)] / hk = [F(xj + (½).hk) - F(xj - (½).hk)] / hk.

Finally, let F(b) = F(b - hk) + f(xk).hk. Then f(xk) = [F(b) - F(b - hk)] / hk = [F(xk + (½).hk) - F(xk - (½).hk)] / hk, and F(b) - F(a) = [f(x1) + ... + f(xk)].hk.

As k increases we get in the limit: F'(x) = f(x), and F(b) - F(a) = Integral(a, b): f(x).dx.

It may be difficult or impossible to find a formula for F(x) when f(x) is given. Known formulas are tabulated in many mathematical handbooks. If there is no formula, the integral must be evaluated numerically.

More Properties of Integrals

Integrals of Products

There is no simple formula for the integral of a product, but the following equation (integration by parts) is often useful:
Integral(a, b): f '(x).g(x).dx + Integral(a, b): f(x).g'(x).dx = f(b).g(b) - f(a).g(a).

Change of Variable

Let x = g(u). Then u = g-1(x), and we have:
Integral(a, b): f(x).dx = Integral(g-1(a), g-1(b)): f [g(u)].g'(u).du.

Differentiation with Respect to a Parameter

Suppose the function f(x, s) to be integrated and the endpoints a(s), b(s) of the integral contain a parameter s. Let I(s) = Integral(a(s), b(s)): f(x, s).dx. Then the derivative of the integral with respect to s is:
I'(s) = Integral(a(s), b(s)): f2(x, s).dx + f(b, s).b'(s) - f(a, s).a'(s),

where f2(x, s) is the partial derivative of f(x, s) with respect to s.

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By R. H. B. Exell, 1998. King Mongkut's University of Technology Thonburi.