# continuous function

**continuous function** A function from one partially ordered set to another having the property, roughly speaking, that least upper bounds are preserved. A function *f *: *S *→ *T*

is said to be continuous if, for every directed subset *X* of *S*, *f* maps the least upper bound of *X* to the least upper bound of the image of *X* under *f*. Continuous functions are significant in denotational semantics since they correspond to the requirement that a computational process produces arbitrarily close approximations to the final output, given arbitrarily close approximations to the total input.

A continuous function *f*(*x*) has no breaks or instantaneous changes in value. In the hierarchy of mathematical functions the smoothest are those, such as sin *x*, cos *x*, that can be differentiated any number of times, always producing a continuous function.

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**continuous function**