Glossary

6.2 Properties of Continuous Functions

5 min read•july 30, 2024

Continuous functions are the building blocks of calculus. They behave predictably, allowing us to analyze their properties and apply them to real-world problems. Understanding these functions is crucial for grasping more complex mathematical concepts.

Properties of continuous functions, like the , help us solve equations and model physical phenomena. We can also combine continuous functions through operations like addition and composition, creating more complex functions that retain continuity.

Properties of Continuous Functions

Sum, Difference, Product, and Quotient of Continuous Functions

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  • If f and g are continuous functions at a point a, then f + g, f - g, f * g, and f / g (provided g(a) ≠ 0) are also continuous at a
    • For example, if f(x) = x^2 and g(x) = sin(x) are continuous at x = π, then f(x) + g(x) = x^2 + sin(x) is also continuous at x = π
  • The sum, difference, product, and quotient of continuous functions are continuous on their respective domains
    • If f(x) = x^3 and g(x) = e^x are continuous on the real numbers, then f(x) - g(x) = x^3 - e^x is also continuous on the real numbers
  • These properties can be used to prove the continuity of more complex functions built from simpler continuous functions
    • The function h(x) = (x^2 + 1) / (x - 1) is continuous on its domain because the numerator and denominator are continuous functions, and the denominator is non-zero on the domain

Intermediate Value Theorem and Applications

  • The Intermediate Value Theorem (IVT) states that if f is continuous on a closed interval [a, b] and f(a) < k < f(b) or f(b) < k < f(a), then there exists a point c in (a, b) such that f(c) = k
    • If f(x) = x^3 - x is continuous on [-1, 1] and f(-1) = -2 < 0 < 2 = f(1), then there exists a point c in (-1, 1) such that f(c) = 0
  • The properties of continuous functions can be applied to solve problems involving limits, derivatives, and integrals
    • To find the limit of a continuous function f(x) as x approaches a, simply evaluate f(a)
    • The derivative of a continuous function is also continuous on its domain
    • Continuous functions on closed intervals are integrable, and the can be applied

Composition of Continuous Functions

Continuity of Composite Functions

  • If f is continuous at a and g is continuous at f(a), then the composite function g ∘ f is continuous at a
    • If f(x) = x^2 is continuous at x = 1 and g(x) = sin(x) is continuous at f(1) = 1, then g ∘ f(x) = sin(x^2) is continuous at x = 1
  • The composition of two continuous functions is continuous on its domain
    • If f(x) = x^3 and g(x) = e^x are continuous on the real numbers, then g ∘ f(x) = e^(x^3) is also continuous on the real numbers

Applications of Composite Function Continuity

  • The continuity of composite functions can be used to prove the continuity of more complex functions
    • The function h(x) = sin(cos(x)) is continuous on the real numbers because sin(x) and cos(x) are continuous functions
  • The chain rule for derivatives and the substitution rule for integrals rely on the continuity of composite functions
    • If f(x) and g(x) are differentiable functions, then the derivative of the composite function g ∘ f(x) is given by (g∘f)′(x)=g′(f(x))⋅f′(x)(g ∘ f)'(x) = g'(f(x)) · f'(x)
    • When evaluating the definite integral ∫abf(g(x))g′(x)dx\int_a^b f(g(x))g'(x)dx, the substitution rule can be applied by setting u=g(x)u = g(x) and du=g′(x)dxdu = g'(x)dx

Continuity of Piecewise Functions

Conditions for Continuity of Piecewise Functions

  • A is continuous if each piece is continuous on its respective domain and the function values agree at the endpoints of adjacent pieces
    • The absolute value function f(x)=∣x∣={x,x≥0−x,x<0f(x) = |x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases} is continuous because both pieces are continuous on their respective domains, and the function values agree at x = 0
  • To determine the continuity of a piecewise-defined function, check the continuity of each piece and the agreement of function values at the endpoints
    • For the function f(x)={x2,x≤12x−1,x>1f(x) = \begin{cases} x^2, & x \leq 1 \\ 2x - 1, & x > 1 \end{cases}, check the continuity of x^2 on (-∞, 1] and 2x - 1 on (1, ∞), and verify that the function values agree at x = 1

Types of Discontinuities in Piecewise Functions

  • Piecewise-defined functions can have removable, jump, or infinite discontinuities at the endpoints of the pieces
    • The function f(x)={x2−1x−1,x≠12,x=1f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1 \\ 2, & x = 1 \end{cases} has a at x = 1
    • The function f(x)={0,x<01,x≥0f(x) = \begin{cases} 0, & x < 0 \\ 1, & x \geq 0 \end{cases} has a at x = 0
    • The function f(x)={1x,x>00,x≤0f(x) = \begin{cases} \frac{1}{x}, & x > 0 \\ 0, & x \leq 0 \end{cases} has an at x = 0
  • Removable discontinuities can be eliminated by redefining the function value at the point of discontinuity to make the function continuous
    • For the function f(x)={x2−1x−1,x≠12,x=1f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1 \\ 2, & x = 1 \end{cases}, redefining f(1) = 2 makes the function continuous at x = 1
  • Jump discontinuities occur when the left-hand and right-hand limits at a point exist but are not equal
    • For the function f(x)={0,x<01,x≥0f(x) = \begin{cases} 0, & x < 0 \\ 1, & x \geq 0 \end{cases}, lim⁡x→0−f(x)=0\lim_{x \to 0^-} f(x) = 0 and lim⁡x→0+f(x)=1\lim_{x \to 0^+} f(x) = 1, resulting in a jump discontinuity at x = 0
  • Infinite discontinuities occur when either the left-hand or right-hand limit at a point is infinite
    • For the function f(x)={1x,x>00,x≤0f(x) = \begin{cases} \frac{1}{x}, & x > 0 \\ 0, & x \leq 0 \end{cases}, lim⁡x→0+f(x)=∞\lim_{x \to 0^+} f(x) = \infty, resulting in an infinite discontinuity at x = 0

Key Terms to Review (18)

Bolzano-Weierstrass Theorem: The Bolzano-Weierstrass Theorem states that every bounded sequence in $b{R}^n$ has a convergent subsequence whose limit lies within the same space. This theorem is fundamental in understanding the properties of sequences and functions, particularly in the context of continuity and optimization.
Boundedness: Boundedness refers to the property of a set or function being contained within specific limits. It means that there exists a number that serves as an upper and lower limit, ensuring that all elements stay within this range. Understanding boundedness is essential for analyzing various mathematical concepts, as it relates to integrability, continuity, and convergence, providing crucial insights into the behavior of functions and sequences.
Compactness: Compactness is a property of a space that essentially combines two key features: being closed and bounded. In a compact space, every open cover has a finite subcover, which means that from any collection of open sets that covers the space, it's possible to select a finite number of those sets that still cover the entire space. This idea is crucial in many areas, as it ensures that certain properties hold true, particularly in relation to continuous functions and optimization.
Connectedness: Connectedness refers to a property of a space in which any two points can be joined by a path that lies entirely within that space. This concept is crucial in understanding how continuous functions behave, as they preserve the connectedness of sets. A space being connected ensures that it cannot be split into two non-empty, disjoint open subsets, providing a foundation for the analysis of continuous mappings and their characteristics.
Continuity at a point: Continuity at a point refers to the property of a function where, at a specific input value, the function's output is defined and approaches the function's value at that input as the input value approaches from either direction. This concept ensures that small changes in the input result in small changes in the output, making the function behave predictably around that point. Understanding continuity at a point is essential for grasping the broader properties of continuous functions, which have implications for limits, derivatives, and integrals.
Continuous Image: A continuous image refers to the result of applying a continuous function to a subset of its domain, which preserves the topological properties of that set. When a continuous function maps a set, the image remains connected and compact if the original set has these properties. This concept is crucial for understanding how functions behave and maintain structure under transformation.
Continuous Mapping: Continuous mapping refers to a function between topological spaces that preserves the notion of closeness; specifically, a function is continuous if the pre-image of every open set is open. This concept is foundational in understanding the properties of continuous functions, as it ensures that small changes in input result in small changes in output, establishing a vital connection in mathematical analysis.
Fundamental theorem of calculus: The fundamental theorem of calculus establishes a deep connection between differentiation and integration, showing that these two operations are essentially inverse processes. It consists of two parts: the first part guarantees that if a function is continuous on an interval, then it has an antiderivative, while the second part provides a method to evaluate definite integrals using antiderivatives. This theorem is pivotal in understanding how integration can be applied to calculate areas and solve real-world problems.
Heine-Borel Theorem: The Heine-Borel Theorem states that a subset of Euclidean space is compact if and only if it is closed and bounded. This theorem is essential in real analysis as it connects the concepts of compactness with two critical properties: closure and boundedness, making it easier to apply results related to continuity and convergence in the analysis of functions.
Infinite Discontinuity: Infinite discontinuity occurs at a point in a function where the function approaches infinity or negative infinity as it gets close to that point. This type of discontinuity indicates that the function does not have a finite limit at that point and is characterized by vertical asymptotes in its graph. Understanding infinite discontinuity is essential because it impacts the properties of integrable functions, the behavior of continuous functions, and the definition of continuity itself.
Intermediate Value Theorem: The Intermediate Value Theorem states that if a function is continuous on a closed interval \\[a, b\\] and takes on different values at the endpoints, then it must take every value between those two values at least once. This theorem is fundamental in understanding the behavior of continuous functions and has deep implications in calculus and analysis.
Jump Discontinuity: A jump discontinuity occurs in a function when there is a sudden 'jump' in the value of the function at a certain point, meaning the left-hand limit and right-hand limit at that point do not match. This type of discontinuity signifies that the function cannot be continuous at that point, as the value of the function does not settle into a single output. Jump discontinuities are crucial for understanding how functions behave in terms of integrability, continuity properties, and how they can be classified in mathematical analysis.
Limit Points: Limit points are values in a topological space that can be approached by a sequence of points from within a given set, meaning every neighborhood of a limit point contains at least one point from that set other than itself. This concept helps in understanding the behavior of sequences and functions near certain points, particularly in relation to continuity. Limit points play a vital role in determining convergence and can indicate the completeness of a set.
Piecewise-Defined Function: A piecewise-defined function is a function that is defined by different expressions or formulas over different intervals of its domain. These functions can capture complex behavior by allowing for various rules to apply in distinct segments, which makes them useful in modeling real-world scenarios and mathematical problems. The continuity and integrability of these functions can vary across the intervals, influencing their properties significantly.
Polynomial functions: Polynomial functions are mathematical expressions that involve variables raised to whole number powers and coefficients that can be real or complex numbers. They take the form $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$, where the coefficients $$a_i$$ are constants, $$n$$ is a non-negative integer, and $$x$$ is the variable. These functions exhibit important properties such as continuity and differentiability, and they play a key role in approximation techniques like Taylor's theorem.
Removable discontinuity: A removable discontinuity occurs at a point in a function where the function is not defined or does not match the limit, but can be 'removed' by redefining the function at that point. This type of discontinuity highlights important aspects of continuity and integrability, as it indicates that while the function may have a gap or break, it could be made continuous by appropriately assigning a value to the discontinuous point.
Trigonometric Functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, primarily used in geometry and analysis. They include sine, cosine, tangent, and their reciprocals, cosecant, secant, and cotangent, and they are fundamental in various applications such as modeling periodic phenomena and analyzing waveforms. These functions are also continuous and periodic, making them essential in calculus and series expansions.
Uniform Continuity: Uniform continuity refers to a stronger form of continuity for functions, where the rate of change of the function is bounded uniformly across its entire domain. This means that for any given tolerance in the output, one can find a corresponding input tolerance that works for all points in the domain, rather than just at individual points. This concept is crucial when considering how functions behave over intervals and plays an important role in various properties of continuity, extreme value considerations, and understanding the completeness of metric spaces.
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