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Continuity definitions

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  • Continuity
    Occurs when approaching a point from both sides yields the same value as the function at that point.
  • Discontinuity
    Arises when a function's limit and value at a point differ, or the limit does not exist.
  • Limit
    Describes the value a function approaches as the input nears a specific point.
  • Hole
    Appears as an undefined point on a graph where the function could be redefined to restore continuity.
  • Jump Discontinuity
    Occurs when the left and right limits at a point differ, causing a sudden change in function value.
  • Asymptote
    Represents a line that a function approaches but never touches, often causing unbounded behavior.
  • Rational Function
    Formed by dividing two polynomials, often exhibiting discontinuities where the denominator is zero.
  • Piecewise Function
    Defined by different expressions over various intervals, possibly causing discontinuities at interval boundaries.
  • One-Sided Limit
    Evaluates the behavior of a function as the input approaches a point from only one direction.
  • Removable Discontinuity
    A type of discontinuity where a hole exists and the function could be redefined to become continuous.
  • Unbounded Behavior
    Describes a function's values increasing or decreasing without limit near a certain point.
  • Function Value
    The output of a function for a specific input, used to compare with the limit for continuity.
  • Graph Tracing
    A visual method for checking continuity by moving a pen along a graph without lifting it.
  • Denominator Zero
    A condition in rational functions that signals potential discontinuities such as holes or asymptotes.
  • Interval Boundary
    A point where different pieces of a piecewise function meet, often requiring special attention for continuity.