Math 2.0

Real functions Definition

Square root function

Builds on Intermediate value theoremStrictly monotone function

Definition

For a real number x≥0x \ge 0, the square root x\sqrt{x} is the unique real number y≥0y \ge 0 with y2=xy^2 = x. The square root function is the function x↦xx \mapsto \sqrt{x} on [0,∞)[0, \infty).

Remarks

The number yy exists by the intermediate value theorem for t↦t2t \mapsto t^2. It is unique because t2t^2 is strictly increasing on [0,∞)[0, \infty). The function is continuous and strictly increasing. It is differentiable on (0,∞)(0, \infty), but not at 00. For x>0x > 0, x=x1/2\sqrt{x} = x^{1/2}. Also x2=∣x∣\sqrt{x^2} = |x| for every real xx. Mathlib defines Real.sqrt, with x=0\sqrt{x} = 0 for x<0x < 0.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionStrictly monotone function, definition, no evidence levelStrictly monotone fun…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionIntermediate value theorem, statement, no evidence levelIntermediate value th…statementSquare root function, definition, no evidence levelSquare root functiondefinition
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  1. Limit point of a set definition
  2. Strictly monotone function definition
  3. Limit of a function definition
  4. Continuity at a point definition
  5. Continuous function definition
  6. Intermediate value theorem statement
  7. Square root function definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Square_root, Properties and uses
About this page
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Kinddefinition
WritingW0 stub
MSC26A09
DefinesSquare root function
UsesIntermediate value theorem, Strictly monotone function

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