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Continuous function

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English: A functin from R to R is continuous iff, informally, its value changes only slightly when its input changes slightly.

Examples

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  • A continuous function
  • A function discontinuous at one point, yet right-continuous there

Counterexamples (discontinuous functions)

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jump discontinuity

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  • A function discontinuous at one point, yet right-continuous there
  • A function discontinuous at one point, yet left-continuous there
  • A function discontinuous at one point
  • Function discontinuous at both points
  • A function discontinuous at several points

Removable discontinuity

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  • A function that has a removable discontinuity at x0 with a point underneath

Continuous from the right with nothing on the left

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  • Square root of x graph. Continuous only from 0 through the right. No points before 0.
  • Xth root of x graph

infinite discontinuity

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  • Tangent of x has an infinite discontinuity at pi/2,-pi/2,3pi/2 and so on - infinite amount of infinite discontinuities
  • 1/x graph 1 infinite discontinuity
  • Rational function with two infinite discontinuities

functions with a finite domain

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  • Inverse sine function graph, y=arcsin(x) , the graph has x values from (-1,1)