2.4 Continuity

4 min readjune 24, 2024

in calculus is all about smooth, unbroken functions. It's the idea that you can draw a function's graph without lifting your pencil. This concept is crucial for understanding limits, derivatives, and integrals.

Continuous functions behave predictably, allowing us to make important calculations and predictions. We'll look at different types of continuity, discontinuities, and how to analyze functions both graphically and algebraically.

Continuity

Continuity at points and intervals

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  • Continuity at a point aa requires three conditions:
    • The of the function as xx approaches aa exists (limxaf(x)\lim_{x \to a} f(x) exists)
    • The function is defined at aa (f(a)f(a) is defined)
    • The limit of the function equals the function value at aa (limxaf(x)=f(a)\lim_{x \to a} f(x) = f(a))
  • Continuity on an interval [a,b][a, b] means:
    • The function is continuous at every point within the interval [a,b][a, b]
    • The function is left-continuous at aa, meaning the left-hand limit equals the function value at aa (limxaf(x)=f(a)\lim_{x \to a^-} f(x) = f(a))
    • The function is right-continuous at bb, meaning the right-hand limit equals the function value at bb (limxb+f(x)=f(b)\lim_{x \to b^+} f(x) = f(b))

Types of discontinuities

  • occurs when:
    • The limit of the function as xx approaches aa exists, but the function is undefined at aa or the function value at aa does not equal the limit (limxaf(x)\lim_{x \to a} f(x) exists, but f(a)f(a) is undefined or f(a)limxaf(x)f(a) \neq \lim_{x \to a} f(x))
    • The discontinuity can be removed by redefining the function value at aa to equal the limit (f(a)=limxaf(x)f(a) = \lim_{x \to a} f(x))
  • happens when:
    • Both one-sided limits exist as xx approaches aa from the left and right, but they are not equal (limxaf(x)\lim_{x \to a^-} f(x) and limxa+f(x)\lim_{x \to a^+} f(x) both exist, but are not equal)
    • The function "jumps" from one value to another at the point of discontinuity
    • This type of discontinuity is common in piecewise functions
  • Infinite discontinuity occurs when:
    • At least one of the one-sided limits is infinite as xx approaches aa from the left or right (limxaf(x)\lim_{x \to a^-} f(x) or limxa+f(x)\lim_{x \to a^+} f(x) is infinite)
    • The function approaches positive or negative infinity near the point of discontinuity
    • This type of discontinuity often involves an asymptote

Applications of intermediate value theorem

  • The (IVT) states:
    • If a function f(x)f(x) is continuous on the closed interval [a,b][a, b] and kk is any value between f(a)f(a) and f(b)f(b), then there exists at least one point cc within the open interval (a,b)(a, b) such that f(c)=kf(c) = k
  • To apply the IVT:
    1. Confirm that the function f(x)f(x) is continuous on the closed interval [a,b][a, b]
    2. Calculate the function values at the endpoints, f(a)f(a) and f(b)f(b)
    3. Determine if the target value kk lies between f(a)f(a) and f(b)f(b)
    4. If all conditions are met, conclude that there exists at least one point cc in the open interval (a,b)(a, b) where f(c)=kf(c) = k

Limits of composite functions

  • The limit of a composite function g(f(x))g(f(x)) as xx approaches aa is:
    • If the limit of the inner function f(x)f(x) as xx approaches aa exists (limxaf(x)=L\lim_{x \to a} f(x) = L) and the limit of the outer function g(x)g(x) as xx approaches LL exists (limxLg(x)=M\lim_{x \to L} g(x) = M), then the limit of the composite function is MM (limxag(f(x))=M\lim_{x \to a} g(f(x)) = M)
    • The limit of the outer function is evaluated at the limit value of the inner function
  • To evaluate the limit of a composite function:
    1. Find the limit of the inner function as xx approaches aa (limxaf(x)\lim_{x \to a} f(x))
    2. If the limit exists, substitute the limit value into the outer function
    3. Evaluate the limit of the outer function at the substituted value

Graphical vs algebraic continuity analysis

  • Graphical method for analyzing continuity:
    • Plot the graph of the function
    • Look for any gaps, jumps, or asymptotes in the graph
    • If the graph is unbroken and has no holes, the function is continuous on its
  • Algebraic method for analyzing continuity:
    • Verify the continuity conditions at each point in the :
      • Find the limit of the function as xx approaches the point from both sides
      • Check if the function is defined at the point
      • Confirm that the limit equals the function value at the point
    • If all three conditions are satisfied at every point in the domain, the function is continuous on its domain

Function properties and continuity

  • Domain and :
    • The domain of a function is the set of all possible input values (x-values)
    • The range is the set of all possible output values (y-values)
    • Continuity is closely related to the domain of a function, as discontinuities often occur at points where the function is undefined
  • Differentiability:
    • If a function is differentiable at a point, it must also be continuous at that point
    • However, a function can be without being differentiable there
    • Differentiability is a stronger condition than continuity

Key Terms to Review (15)

Constant multiple law for limits: The Constant Multiple Law for limits states that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function. Mathematically, if $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L$ where $k$ is a constant.
Continuity: Continuity is a fundamental concept in calculus that describes the smoothness and uninterrupted nature of a function. It is a crucial property that allows for the application of calculus techniques and the study of limits, derivatives, and integrals.
Continuity over an interval: Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
Continuous at a point: A function $f(x)$ is continuous at a point $x = c$ if the limit of $f(x)$ as $x$ approaches $c$ exists, equals $f(c)$, and $f(c)$ is defined.
Continuous from the left: A function is continuous from the left at a point $a$ if the limit of the function as $x$ approaches $a$ from the left exists and equals the function's value at $a$. Mathematically, this is expressed as $\lim_{{x \to a^-}} f(x) = f(a)$.
Continuous from the right: A function $f(x)$ is continuous from the right at $x = c$ if $\lim_{{x \to c^+}} f(x) = f(c)$. This means that as $x$ approaches $c$ from values greater than $c$, the function value approaches $f(c)$.
Discontinuous at a point: A function is discontinuous at a point if there is a sudden jump, break, or hole at that point in its graph. The function does not have a well-defined limit or the value of the function does not match the limit at that point.
Domain: The domain of a function is the set of all possible input values (typically $x$-values) for which the function is defined. It represents all the values that can be plugged into the function without causing any undefined behavior.
Domain: The domain of a function refers to the set of all possible input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is a crucial concept in understanding the behavior and properties of functions.
Intermediate Value Theorem: The Intermediate Value Theorem states that for any continuous function $f$ on a closed interval $[a, b]$, if $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one point $c$ in the interval $(a, b)$ such that $f(c) = N$. This theorem guarantees the existence of a solution within the interval under specific conditions.
Jump discontinuity: A jump discontinuity occurs at a point where the left-hand and right-hand limits of a function exist but are not equal. This results in a 'jump' in the graph of the function at that point.
Limit: In mathematics, the limit of a function is a fundamental concept that describes the behavior of a function as its input approaches a particular value. It is a crucial notion that underpins the foundations of calculus and serves as a building block for understanding more advanced topics in the field.
Piecewise function: A piecewise function is a function that is defined by multiple sub-functions, each applying to a specific interval or condition within its domain. This type of function allows for different formulas to govern different parts of the input values, making it versatile for modeling real-world scenarios where a single formula might not be adequate. Piecewise functions are essential for understanding how functions behave differently across various segments, and they are often analyzed in terms of their continuity at the boundaries between these segments.
Range: Range refers to the set of all possible output values (or 'y' values) that a function can produce based on its domain (the set of input values). Understanding the range helps us grasp how a function behaves, what outputs are attainable, and the limitations on those outputs.
Removable discontinuity: A removable discontinuity occurs when a function has a hole at a particular point, which can be 'fixed' by redefining the function at that point. This happens when the limit of the function as it approaches the point exists but is not equal to the function's value at that point.
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