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Continuity

Continuity. What is Continuity?. Geometrically, this means that there is NO gap, split, or missing pt. (hole) for f(x) at c. A pencil could be moved along the graph of f(x) through (c, f(c)) WITHOUT lifting it off of the graph.

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Continuity

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  1. Continuity

  2. What is Continuity? • Geometrically, this means that there is NO gap, split, or missing pt. (hole) for f(x) at c. • A pencil could be moved along the graph of f(x) through (c, f(c)) WITHOUT lifting it off of the graph. • The function not only intended to reach a certain height (limit) but it actually did: Limit exists + Fnc. Defined = Continuity • Discontinuity occurs when there is a hole in the graph even if the graph doesn’t actually break into 2 different pieces.

  3. Formal Definition of Continuity • f(c) exists (c is in the domain of f) • lim f(x) exists • lim f(x) = f(c) *NOTE: When a fnc. increases and decreases w/o bound around a vertical asymptote (x=c), then the fnc. demonstrates infinite discontinuity. A function is continuous at a point if the limit is the same as the value of the function.

  4. 2 1 1 2 3 4 Examples This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

  5. Types of Discontinuities: • Removable Discontinuities: • Essential Discontinuities: Jump Infinite Oscillating (You can fill the hole.)

  6. has a discontinuity at x=1. Removing a Discontinuity: Write an extended function that is continuous at x=1.

  7. Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. • Also, composites of continuous functions are continuous. *NOTE: Graphing calculators can make non-continuous functions appear continuous; the calculator “connects the dots” which covers up the discontinuities.

  8. Evaluate Continuity at the given pt.: • f(x) = 2x+3 at x = -4 • f(x) = at x = 2 • f(x) = at x = 0

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