12.3 Continuity

Continuity 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 limit 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 $x=a$
  • $f(a)$ must be defined meaning the function exists at $x=a$ (no gaps or holes)
  • $\lim_{x \to a} f(x)$ must exist meaning the limit of the function as $x$ approaches $a$ from both sides converges to a single value
  • $\lim_{x \to a} f(x) = f(a)$ meaning the limit of the function as $x$ approaches $a$ equals the function value at $x=a$ (no jumps or breaks)
• Identifying points of continuity and discontinuity involves checking the three continuity conditions at each point of interest
  • Pay special attention to domain restrictions 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, step functions)
  • Consider one-sided continuity when examining functions defined on intervals with endpoints

Evaluating continuity at specific values

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

Types of function discontinuities

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

Continuity and Differentiability

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