Continuity: Fresh Take

  • Outline the three criteria a function must meet to be continuous at a specific point
  • Explain the different types of breaks a function can have that make it not continuous
  • Explain what it means for a function to be continuous over a range of values
  • Explain the rule for calculating limits of functions that are combined
  • Show how a continuous function reaches every value between its start and end points using the Intermediate Value Theorem

Continuity

The Main Idea 

  • A function [latex]f(x)[/latex] is continuous at [latex]x = a[/latex] if:
    • f(a) is defined
    • [latex]\lim_{x \to a} f(x)[/latex] exists
    • [latex]\lim_{x \to a} f(x) = f(a)[/latex]
  • Types of Discontinuities:
    • Removable (hole in the graph)
    • Jump (function value jumps at a point)
    • Infinite (function approaches infinity near a point)
  • Continuity of Functions:
    • Polynomials are continuous everywhere
    • Rational functions are continuous except where denominator is zero
  • Analyzing Continuity:
    • Check if function is defined at the point
    • Evaluate the limit as [latex]x[/latex] approaches the point
    • Compare the limit value with the function value at the point

Using the definition, determine whether the function [latex]f(x)=\begin{cases} 2x+1 & \text{ if } \, x < 1 \\ 2 & \text{ if } \, x = 1 \\ -x+4 & \text{ if } \, x > 1 \end{cases}[/latex] is continuous at [latex]x=1[/latex].

If the function is not continuous at [latex]1[/latex], indicate the condition for continuity at a point that fails to hold.

For what values of [latex]x[/latex] is [latex]f(x)=3x^4-4x^2[/latex] continuous?

Types of Discontinuities

The Main Idea 

  • Three Main Types of Discontinuities:
    • Removable (hole in the graph)
    • Jump (function value jumps at a point)
    • Infinite (function approaches infinity near a point)
  • Removable Discontinuity:
    • Limit exists at the point, but function value is different or undefined
    • Graphically appears as a hole in the function
  • Jump Discontinuity:
    • Left-hand and right-hand limits exist but are not equal
    • Function “jumps” from one value to another
  • Infinite Discontinuity:
    • Function approaches infinity as it nears the point
    • Often associated with vertical asymptotes
  • Identifying Discontinuities:
    • Evaluate function at the point
    • Calculate left-hand and right-hand limits
    • Compare limits to function value

For [latex]f(x)=\begin{cases} x^2 & \text{ if } \, x \ne 1 \\ 3 & \text{ if } \, x = 1 \end{cases}[/latex], decide whether [latex]f[/latex] is continuous at [latex]1[/latex].

If [latex]f[/latex] is not continuous at [latex]1[/latex], classify the discontinuity as removable, jump, or infinite.

State the interval(s) over which the function [latex]f(x)=\sqrt{x+3}[/latex] is continuous.

Continuity Over an Interval

The Main Idea 

  • Continuity on Open Intervals:
    • Function is continuous at every point within [latex](a, b)[/latex]
  • Continuity on Closed Intervals:
    • Function is continuous on [latex](a, b)[/latex]
    • Right-continuous at [latex]a[/latex]: [latex]\lim_{x \to a^+} f(x) = f(a)[/latex]
    • Left-continuous at [latex]b[/latex]: [latex]\lim_{x \to b^-} f(x) = f(b)[/latex]
  • Half-Open Intervals:
    • For [latex](a, b][/latex]: Continuous on [latex](a, b)[/latex] and left-continuous at [latex]b[/latex]
    • For [a, b): Continuous on [latex](a, b)[/latex] and right-continuous at [latex]a[/latex]
  • Determining Continuity:
    • Check domain of the function
    • Analyze behavior at endpoint(s) for closed intervals
    • Consider discontinuities within the interval

Determine the interval(s) of continuity for the function:

[latex]f(x) = \begin{cases} \sqrt{x+1} & \text{if } x < 3 \\ \frac{x-3}{x-2} & \text{if } x \geq 3 \end{cases}[/latex]

Composite Function Theorem and The Intermediate Value Theorem

The Main Idea 

  • Composite Function Theorem:
    • If [latex]f(x)[/latex] is continuous at [latex]L[/latex] and [latex]\lim_{x \to a} g(x) = L[/latex], then: [latex]\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)) = f(L)[/latex]
    • Helps expand our ability to compute limits
    • Demonstrates continuity of trigonometric functions
  • Continuity of Trigonometric Functions:
    • All trigonometric functions are continuous over their entire domains
    • Proof uses Composite Function Theorem and continuity of [latex]\sin x[/latex] and [latex]\cos x[/latex] at [latex]0[/latex]
  • Intermediate Value Theorem (IVT):
    • Applies to functions continuous over closed, bounded intervals [latex][a,b][/latex]
    • If [latex]z[/latex] is between [latex]f(a)[/latex] and [latex]f(b)[/latex], there exists [latex]c[/latex] in [latex][a,b][/latex] where [latex]f(c) = z[/latex]
    • Useful for proving existence of solutions (e.g., zeros of functions)
  • Applications of IVT:
    • Proving existence of zeros for continuous functions
    • Cannot be used to prove non-existence of zeros or other values

For [latex]f(x)=1/x, \, f(-1)=-1<0[/latex] and [latex]f(1)=1>0[/latex]. Can we conclude that [latex]f(x)[/latex] has a zero in the interval [latex][-1,1][/latex]?