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Continuity and Discontinuities

Sep 10, 2025

Overview

This lecture explores the concept of continuity in functions, different types of discontinuities, properties of continuous functions, and the Intermediate Value Theorem (IVT). The instructor uses examples and problem-solving to illustrate these ideas and clarify common misconceptions.

Warmup Problem: Limits and Composition

  • The statement "If limₓ→ₐ f(x) = b and lim_y→_b g(y) = L, then limₓ→ₐ g(f(x)) = L" is false in general.
    • Intuitively, it may seem true, but a counterexample shows it can fail if g is not continuous at b.
    • Example: If f(x) and g(x) are both defined as 0 for x ≠ 0 and 1 for x = 0, with a = b = 0, then limₓ→₀ f(x) = 0 and lim_y→₀ g(y) = 0, but limₓ→₀ g(f(x)) = 1, not 0.
    • The failure occurs because the composition can "hit" the discontinuity at b, even if the limits of the individual functions exist.

Continuity and Discontinuities

  • A function is continuous at a point if all three of the following hold:
    1. The limit as x approaches the point exists.
    2. The function is defined at that point.
    3. The limit equals the function value at that point.
  • Removable Discontinuity:
    • The limit exists, but the function value at the point is different from the limit.
    • Can be "fixed" by redefining the function at that point to match the limit.
    • Example: For f(x) = x + 3 except at x = 1 (where f(1) = 2), the limit as x→1 is 4. Redefining f(1) = 4 makes the function continuous at x = 1.
  • Jump Discontinuity:
    • The left and right limits at a point both exist but are not equal.
    • Cannot be fixed by redefining the function at that point.
    • Example: A piecewise function with different expressions on either side of a point, where the limits from each side do not match.
  • Other Discontinuities:
    • Occur when the limit does not exist at a point, such as when the function blows up to infinity or oscillates infinitely.
    • Example: h(x) = -1/x for x < 0, h(x) = sin(1/x) for x > 0, h(0) = 2. The limit as x→0 does not exist due to infinite oscillation and blowup.

Making Piecewise Functions Continuous

  • To make a piecewise function continuous, ensure the limits from both sides at each "gluing" point are equal and match the function value (if defined).
  • Example: For a function defined as 3x² for x < -1, ax + b for -1 ≤ x ≤ 1, and 4x² + 1 for x > 1:
    • Set up equations so that the value from the left and right at x = -1 and x = 1 match.
    • Solve for a and b so that the function is continuous at both points.
    • The process involves equating the limits from each side and solving the resulting system of equations.

Examples of Continuous Functions

  • Polynomials: Always continuous everywhere.
  • Rational functions: Continuous wherever the denominator is not zero.
  • Other common continuous functions: eˣ, sin(x), cos(x), arctan(x), √x (for x ≥ 0), log(x) (for x > 0), tangent (where defined).
  • Combining continuous functions:
    • The sum, difference, product, and (where the denominator is nonzero) quotient of continuous functions are continuous.
    • The composition of continuous functions is also continuous.

Limits and Continuous Functions

  • For a function continuous at a point, the limit as x approaches that point equals the function value.
    • This means you can simply substitute the value into the function to find the limit.
    • Example: For a complicated function built from continuous pieces, the limit as x approaches a value can be found by direct substitution.

Intermediate Value Theorem (IVT)

  • Statement: If a function is continuous on [a, b], then for any value t between f(a) and f(b), there exists some c in (a, b) such that f(c) = t.
    • The function must attain every value between f(a) and f(b) at least once.
    • The function may go outside the interval [f(a), f(b)] elsewhere, but it must hit all intermediate values.
    • The IVT is often described as the "no lifting your pencil" property: you cannot jump over any value between f(a) and f(b) without passing through it.

IVT: Problem Examples

  • Polynomials and Roots:
    • If a polynomial (or any continuous function) changes sign between two points, it must have a root in that interval.
    • Example: For P(x) = 18x³ - 63x² + 67x - 20, P(0) = -20 and P(1) = 2. Since the function goes from negative to positive, IVT guarantees a root between 0 and 1.
    • If the function does not change sign, IVT does not guarantee a root, but the function could still cross zero if it dips below and comes back up.
  • Cos(x) = x Example:
    • To show there is a solution to cos(x) = x between 0 and 1, consider f(x) = cos(x) - x.
    • f(0) = 1 (positive), f(1) = cos(1) - 1 (negative).
    • By IVT, there must be some c in (0, 1) where f(c) = 0, i.e., cos(c) = c.

Key Terms & Definitions

  • Continuous Function: A function with no jumps, holes, or breaks at a point or on an interval.
  • Removable Discontinuity: A point where the limit exists but does not equal the function value; can be fixed by redefining the function at that point.
  • Jump Discontinuity: A point where the left and right limits exist but are not equal; cannot be fixed by redefining the function at that point.
  • Intermediate Value Theorem: States that a continuous function on [a, b] attains every value between f(a) and f(b).

Action Items / Next Steps

  • Practice identifying and classifying types of discontinuities in various functions.
  • Work on problems involving making piecewise functions continuous by matching limits at endpoints.
  • Review the properties of continuous functions and how to combine them.
  • Apply the Intermediate Value Theorem to determine the existence of roots and other values in continuous functions.

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