Discontinuous or Continuous

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What is a continuous function?

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A continuous function is a type of function where small changes in the input result in small changes in the output. It can be graphed without lifting the pencil from the paper.

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What is a discontinuous function?

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A discontinuous function is a function that has one or more points at which it is not continuous. This means there are jumps, breaks, or holes in the graph.

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Give an example of a continuous function.

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An example of a continuous function is f(x) = x^2, which is a parabola that can be drawn without lifting the pencil.

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Give an example of a discontinuous function.

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An example of a discontinuous function is f(x) = 1/x, which is not defined at x = 0, creating a break in the graph.

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What does it mean for a function to be defined at a point?

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A function is defined at a point if there is a corresponding output value for that input value.

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What is a jump discontinuity?

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A jump discontinuity occurs when the left-hand limit and the right-hand limit at a certain point are not equal, causing a 'jump' in the graph.

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What is an infinite discontinuity?

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An infinite discontinuity occurs when the function approaches infinity or negative infinity as it approaches a certain input value.

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What is a removable discontinuity?

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A removable discontinuity occurs when a function is not defined at a point, but can be made continuous by defining it at that point.

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How can you determine if a function is continuous at a point?

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A function is continuous at a point if the limit as it approaches that point equals the function's value at that point.

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What is the Intermediate Value Theorem?

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The Intermediate Value Theorem states that if a function is continuous on a closed interval, then it takes on every value between its values at the endpoints of the interval.