Glossary

12.3 Continuity

3 min read•june 25, 2024

in functions is all about smooth, unbroken behavior. It's like drawing a line without lifting your pencil - no gaps, jumps, or breaks allowed. Understanding continuity helps us predict how functions behave and where they might have interesting quirks.

To be continuous, a function needs to exist at a point, have a as we approach that point, and have the limit match the function's value. We'll explore different types of discontinuities and how to spot them, which is crucial for analyzing function behavior.

Continuity

Points of continuity and discontinuity

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  • Continuity requires a function to satisfy three conditions at a point x=ax=a
    • f(a)f(a) must be defined meaning the function exists at x=ax=a (no gaps or holes)
    • lim⁡x→af(x)\lim_{x \to a} f(x) must exist meaning the limit of the function as xx approaches aa from both sides converges to a single value
    • lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a) meaning the limit of the function as xx approaches aa equals the function value at x=ax=a (no jumps or breaks)
  • Identifying and involves checking the three continuity conditions at each point of interest
    • Pay special attention to that can cause discontinuities (division by zero, even roots of negatives)
    • Analyze the function's behavior around points where the formula changes or has a piecewise definition (absolute value, )
    • Consider when examining functions defined on intervals with endpoints

Evaluating continuity at specific values

  • Determining if a function is continuous at a specific point x=ax=a involves a step-by-step process
    • Check if f(a)f(a) is defined by plugging aa into the function (ensure aa is in the domain)
    • Evaluate the limit of the function as xx approaches aa from both the left and right sides
      • If the left-hand and right-hand limits both exist and are equal, the overall limit exists ()
    • Compare the overall limit (if it exists) to the function value f(a)f(a)
      • If the limit equals the function value, the function is continuous at x=ax=a ()
      • If the limit does not exist or does not equal the function value, the function is discontinuous at x=ax=a (fails continuity test)
  • The provides a more rigorous approach to proving continuity at a point

Types of function discontinuities

  • () occurs when the limit exists at x=ax=a, but either f(a)f(a) is undefined or does not equal the limit
    • Can be "removed" by redefining f(a)f(a) to match the limit value (filling in a hole)
    • Example: f(x)=x2−1x−1f(x) = \frac{x^2-1}{x-1} has a removable discontinuity at x=1x=1 because lim⁡x→1f(x)=2\lim_{x \to 1} f(x) = 2 but f(1)f(1) is undefined
  • occurs when the left and right limits at x=ax=a both exist but are not equal
    • The function "jumps" from one value to another at the point of discontinuity (step functions)
    • Example: f(x)={1,x<02,x≥0f(x) = \begin{cases} 1, & x < 0 \\ 2, & x \geq 0 \end{cases} has a jump discontinuity at x=0x=0 because lim⁡x→0−f(x)=1\lim_{x \to 0^-} f(x) = 1 and lim⁡x→0+f(x)=2\lim_{x \to 0^+} f(x) = 2
  • () occurs when the limit as xx approaches aa from either or both sides is infinite
    • The function values approach positive or negative infinity near x=ax=a (rational functions)
    • Example: f(x)=1xf(x) = \frac{1}{x} has an infinite discontinuity at x=0x=0 because lim⁡x→0f(x)=±∞\lim_{x \to 0} f(x) = \pm\infty
  • occurs when the function oscillates between multiple values near x=ax=a without approaching a single limit
    • The left and right limits do not exist due to the oscillating behavior (trigonometric functions)
    • Example: f(x)=sin⁡(1x)f(x) = \sin(\frac{1}{x}) has an oscillating discontinuity at x=0x=0 because the function oscillates more rapidly between -1 and 1 as xx approaches 0

Continuity and Differentiability

  • is a prerequisite for
  • is a stronger condition than continuity, ensuring consistent behavior across an entire interval
  • Differentiability implies continuity, but the converse is not always true

Key Terms to Review (27)

|x - a| < δ: |x - a| < δ is a mathematical expression that defines the concept of continuity. It represents the idea that a function is continuous at a point 'a' if the distance between the function's value at 'x' and the function's value at 'a' can be made arbitrarily small by choosing 'x' sufficiently close to 'a'. This is a crucial concept in the study of calculus and the analysis of the behavior of functions.
Continuity: Continuity is a fundamental concept in mathematics that describes the smooth and uninterrupted behavior of a function. It is a property that ensures a function's values change gradually without any sudden jumps or breaks, making it a crucial consideration in the study of various mathematical functions and their applications.
Continuity on an Interval: Continuity on an interval refers to the property of a function where the function's values change in a smooth, uninterrupted manner over a specified range of input values. This means that the function has no abrupt jumps or breaks within the given interval, and the function's graph can be drawn without lifting the pencil from the paper.
Continuous Function: A continuous function is a mathematical function where the output value changes smoothly and without any abrupt jumps or breaks as the input value is varied. This concept of continuity is crucial in the study of exponential functions and the analysis of the continuity of a function.
Differentiability: Differentiability is a fundamental concept in calculus that describes the smoothness and continuity of a function. It is the property of a function that allows for the existence of a derivative, which represents the rate of change of the function at a given point.
Discontinuity: Discontinuity refers to a break or interruption in the continuity or smoothness of a function. It is a point where a function is not defined or does not have a well-defined value, causing a gap or jump in the graph of the function.
Domain Restrictions: Domain restrictions refer to the limits or constraints placed on the input values, or domain, of a function. This concept is crucial in understanding the behavior and properties of various functions, including rational functions, inverse trigonometric functions, and the continuity of functions.
Epsilon-Delta Definition: The epsilon-delta definition is a precise mathematical way of defining the concept of a limit. It provides a rigorous framework for determining whether a function approaches a specific value as the input approaches a particular point. This definition is crucial in the context of finding limits and understanding continuity in calculus.
Extreme Value Theorem: The Extreme Value Theorem states that a continuous function on a closed interval must have both a maximum and a minimum value within that interval. This means that if a function is continuous on a closed interval, it will have at least one absolute maximum and one absolute minimum value within that interval.
Infinite Discontinuity: An infinite discontinuity occurs when a function has a vertical asymptote, meaning the function approaches positive or negative infinity at a particular point. This type of discontinuity is considered 'infinite' because the function's value cannot be defined at that point, as it would result in division by zero, which is undefined.
Intermediate Value Theorem: The Intermediate Value Theorem states that if a continuous function takes on two different values, then it must also take on every value in between those two values. This theorem is a fundamental concept in calculus and has important implications in the study of polynomial functions, limits, and continuity.
Jump Discontinuity: A jump discontinuity occurs when a function experiences a sudden, abrupt change in value at a specific point, resulting in a discontinuous graph. This term is particularly relevant in the context of finding limits and understanding the concept of continuity in mathematics.
Left-Hand Limit: The left-hand limit of a function at a point is the value that the function approaches as the input variable approaches the point from the left. It represents the behavior of the function as it approaches the point from the left side.
Lim f(x) = L: The limit of a function f(x) as x approaches a particular value is equal to the constant L. This means that as x gets closer and closer to a specific value, the function f(x) gets arbitrarily close to the value L. The limit represents the behavior of the function as it approaches a particular point, even if the function is not defined at that point.
Limit: A limit is a value that a function or sequence approaches as the input approaches a particular point or as the index approaches infinity. It represents the behavior of a function or sequence at a specific point or in the long run.
Limit Definition of Continuity: The limit definition of continuity describes a function as continuous at a point if the function's limit at that point equals the function's value at that point. This means that as the input values approach a specific point, the function's output values also approach the function's value at that point, without any sudden jumps or breaks in the function's graph.
One-Sided Continuity: One-sided continuity refers to the concept of a function being continuous from one side, either from the left or the right, at a particular point. This is an important consideration in the study of continuity, as a function may exhibit different behavior on either side of a point.
Oscillating Discontinuity: An oscillating discontinuity is a type of discontinuity in a function where the function value oscillates, or fluctuates, around a specific point. This discontinuity occurs when the left-hand and right-hand limits of the function do not exist at a particular point, causing the function to 'jump' or 'oscillate' at that point.
Piecewise Function: A piecewise function is a function that is defined by different formulas or expressions over different intervals or domains of the independent variable. These functions are composed of multiple parts, each with its own rule or equation, allowing for greater flexibility in modeling real-world phenomena.
Point Discontinuity: A point discontinuity refers to a specific type of discontinuity in a function where the function is not defined at a particular point, causing a break or gap in the graph. This type of discontinuity is often characterized by the function's value not existing or being undefined at a specific x-value.
Points of Continuity: Points of continuity refer to the specific locations on a function's graph where the function is continuous. Continuity is a fundamental property in calculus that describes the smooth and uninterrupted behavior of a function at a given point.
Removable Discontinuity: A removable discontinuity is a point where a function is not defined, but the function can be made continuous by assigning a value at that point. It is a type of discontinuity that can be 'removed' by redefining the function to be continuous at that point.
Right-Hand Limit: The right-hand limit of a function at a particular point is the value that the function approaches as the input variable approaches the point from the right side. It represents the behavior of the function as it approaches the point from the right, independent of the function's behavior from the left side.
Sandwich Theorem: The Sandwich Theorem, also known as the Squeeze Theorem or the Pinching Theorem, is a fundamental result in real analysis that provides a sufficient condition for the existence of the limit of a function. It states that if a function is bounded between two other functions that both converge to the same limit, then the original function must also converge to that same limit.
Step Functions: A step function is a mathematical function that changes value abruptly at certain points, known as discontinuities, and remains constant between these points. This type of function is often used to model real-world phenomena that exhibit discrete or sudden changes, such as the pricing of certain goods or the operation of electrical circuits.
Uniform Continuity: Uniform continuity is a stronger form of continuity that ensures a function behaves consistently across its entire domain. It guarantees that the function's values change by an arbitrarily small amount whenever the input values change by a sufficiently small amount, regardless of the specific location within the function's domain.
Vertical Asymptote: A vertical asymptote is a vertical line that a graph of a function approaches but never touches. It represents the vertical limit of a function's behavior, indicating where the function becomes undefined or experiences a discontinuity.
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