pathological

In mathematics, a pathological object is mathematicalobject that has a highly unexpected .

Pathological objects are typically percieved to, in some sense, bebadly behaving. On the other hand, they are perfectly properlydefined mathematical objects. Therefore this “bad behaviour” cansimply be seen as a contradiction with our intuitivepicture of how a certain object should behave.

Examples

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A very famous pathological function is theWeierstrass function, which is a continuous functionthat is nowhere differentiable.
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The Peano space filling curve. This pathological curvemaps the unit interval $[0,1]$ continuously onto $[0,1]\\times[0,1]$ .
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The Cantor set. This is subset of the interval $[0,1]$ has the pathological property that it is uncountableyet its measure is zero.
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The Dirichlet’s function from $\\mathbbmss{R}$ to $\\mathbbmss{R}$ is continuous at everyirrational point and discontinuous at every rational point.
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Ackermann Function.

References

1 Wikipedia http://en.wikipedia.org/wiki/Pathological (mathematics)entry on pathological, mathematics.

单词	Pathological
释义	pathological In mathematics, a pathological object is mathematicalobject that has a highly unexpected . Pathological objects are typically percieved to, in some sense, bebadly behaving. On the other hand, they are perfectly properlydefined mathematical objects. Therefore this “bad behaviour” cansimply be seen as a contradiction with our intuitivepicture of how a certain object should behave. Examples • A very famous pathological function is theWeierstrass function, which is a continuous functionthat is nowhere differentiable. • The Peano space filling curve. This pathological curvemaps the unit interval $[0,1]$ continuously onto $[0,1]\\times[0,1]$ . • The Cantor set. This is subset of the interval $[0,1]$ has the pathological property that it is uncountableyet its measure is zero. • The Dirichlet’s function from $\\mathbbmss{R}$ to $\\mathbbmss{R}$ is continuous at everyirrational point and discontinuous at every rational point. • Ackermann Function. See also [1]. References 1 Wikipedia http://en.wikipedia.org/wiki/Pathological (mathematics)entry on pathological, mathematics.
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