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Continuity definitions

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  • Continuity
    A property where the limit at a point matches the function value, allowing smooth tracing through that point on a graph.
  • Discontinuity
    A break in a function where the limit and function value differ, often seen as holes, jumps, or asymptotes on a graph.
  • Limit
    The value a function approaches as the input nears a specific point, crucial for determining continuity.
  • Function Value
    The output of a function at a specific input, compared with the limit to assess continuity.
  • Hole
    A type of discontinuity where the function is undefined at a point, but the limit exists.
  • Jump Discontinuity
    A sudden change in function value at a point, where left and right limits differ.
  • Asymptote
    A line that a function approaches but never touches, often causing unbounded behavior and discontinuity.
  • Rational Function
    A function formed by dividing two polynomials, often exhibiting holes or asymptotes.
  • Piecewise Function
    A function defined by different expressions over different intervals, often leading to jumps at boundaries.
  • One-Sided Limit
    The value a function approaches from only one direction (left or right) as the input nears a point.
  • Removable Discontinuity
    A hole in a function's graph that could be 'filled' to make the function continuous at that point.
  • Unbounded Behavior
    A situation where function values increase or decrease without limit near a point, often linked to asymptotes.
  • Denominator
    The bottom part of a rational function, whose zeros often indicate locations of discontinuity.
  • Graph Tracing
    A visual method for checking continuity by seeing if a function can be drawn through a point without lifting the pen.
  • Boundary Point
    A value separating pieces in a piecewise function, often checked for continuity using one-sided limits.