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This section deals with continuity, (Is this function continuous?), discontinuities (removable, jump/step, infinite), and the intermediate value theorem.
- Removable discontinuities are holes.
- Non-removable discontinuities are infinite discontinuities or step/jump discontinuities.
- The Intermediate Value Theorem (or IVT) simply says that if you connect two points with a continuous function, it must pass through all of the y values (outputs) between the two points.
Danger zones for continuity:
1. Holes. Holes are removable discontinuities and will always give you an x value where the function is discontinuous.
2. Negatives in square roots. Find the range of values that would give you a negative in the square root to determine where a function is discontinuous.
3. Negatives or zeroes in logarithms. Find the range of values that would give you a negative or a zero in a logarithm to determine where a function is discontinuous.
4. The intersections of graphs in piece-wise functions: In piece-wise functions, the added layer of complexity comes from the intersections between the different parts of the piece-wise functions. If the limits don't match on both sides (see here for more on this), then the function is discontinuous at that point.
5. Vertical Asymptotes / Infinite Discontinuities Any x values that make the bottom of a fraction zero are vertical asymptotes and you must show that those function are discontinuous at those points.
Practice worksheet: http://cdn.kutasoftware.com/Worksheets/Calc/02%20-%20Continuity.pdf
See here for more on continuity: http://17calculus.com/limits/continuity/
See here for IVT: http://en.wikipedia.org/wiki/Intermediate_value_theorem
Example Problem:
More video links:
Continuity:
Intermediate Value Theorem:
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