Undergraduate → Real Analysis ↓
Functions of Real Variables
In real analysis, functions of a real variable are the fundamental objects of study. A function is a relation that associates each element of one set, known as the domain, with an element of another set, called the codomain. When considering a function of a real variable, both the domain and the codomain are subsets of the real numbers R
Basic definitions
The function f defined on the set D of real numbers is expressed as:
f: D → RThere is a unique real number y = f(x) for every x ∈ D Here, D is the domain, and the set of all outputs f(x) is the range.
Example: linear function
Consider a simple linear function:
f(x) = 2x + 3For each real number x, the function outputs 2x + 3 So, for x = 2, f(2) = 2(2) + 3 = 7.
Polynomial function
Polynomial functions are another important example of functions of a real variable. It is of the following form:
f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0where a_0, a_1, ..., a_n are real numbers, and n is a non-negative integer. A special type of polynomial function is the quadratic function.
Example: Quadratic function
Consider the quadratic function:
f(x) = x^2 - 4x + 4This function is represented by a parabola on the graph.
Continuous work
A function f is continuous at a point c if for every positive number ε, there exists a positive number δ such that if |x - c| < δ, then |f(x) - f(c)| < ε Intuitively, the graph of a continuous function can be drawn without lifting the pencil from the paper.
Example: Continuous function
For the function f(x) = x^3, it is continuous for all real numbers.
Discontinuous functions
A function f is discontinuous at a point c if it is not continuous at c. This means that there may be "jumps" or "gaps" at some points.
Example: Step function
The step function is a classic example of a discontinuous function:
f(x) = { 1 if x >= 0, 0 if x < 0 }Bounded function
A function f is said to be bounded if there exist real numbers M and m such that for all x in the domain, m ≤ f(x) ≤ M
Example: Sine function
An example of a bounded function is the sine function, f(x) = sin(x) It is bounded because for all real numbers x, -1 ≤ sin(x) ≤ 1.
Monotonic functions
A function f is called monotonic if it is absolutely non-increasing or non-decreasing on its domain. If f is non-decreasing, then for x_1 ≤ x_2, we have f(x_1) ≤ f(x_2) If f is non-increasing, then for x_1 ≤ x_2, we have f(x_1) ≥ f(x_2).
Example: Exponential function
The exponential function f(x) = e^x is an example of a monotonic increasing function.
Inverse function
The inverse function essentially reverses the roles of the input and output. If f is a function from A to B, then the inverse function f -1 is a function from B to A such that for each y ∈ B, f -1 (y) = x if and only if f(x) = y.
Example: Logarithmic function
Consider the exponential function f(x) = e^x. Its inverse is the natural logarithm function f -1 (x) = ln(x).
The graphs of f(x) = e^x and f -1 (x) = ln(x) are symmetric with respect to the line y = x.
Closing thoughts
Functions of real variables form the cornerstone of real analysis and serve as essential tools in a variety of disciplines, including mathematics, engineering, and science. Whether continuous, monotonic, bounded, or inverse, each function describes a unique relationship between inputs and outputs. Understanding these concepts is crucial for analyzing and interpreting mathematical relationships involving real numbers.
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