Continuity definitions Flashcards
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.
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