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We look at continuity for maps between metric spaces .

**Definition**

A map *f* between metric spaces is **continuous at a point p X** if

Given > 0 > 0 such that

Informally: points close to *p* (in the metric *d*_{X}) are mapped close to *f*(*p*) (in the metric *d*_{Y}).

A *continuous function* is one which is continuous for all *p* *X*.

**Remarks**

When one is given a point *p* and > 0 the one needs for the definition may depend on both *p* and . It is therefore *incorrect* to define continuity as:

*p* *X*, > 0 > 0 such that *x* *X* with *d*_{X}(*p*, *x*) < *d*_{X}(*f*(*p*), *f*(*x*)) < .

since this would imply that the same choice of would work for all *p*.

As in the prototype **R** case, one can connect continuity and convergence with:

**Theorem**

*Continuous functions map convergent sequences to convergent sequences. *

Formally if *f*: *X* *Y* is a map between metric spaces which is continuous and (*a*_{n}) is a sequence in *X* which is

convergent to a point *p* *X* them (*f*(*a*_{n})) is a sequence in *Y* convergent to *f*(*p*).

**Proof**

Points close to *p* are mapped close to *f*(*p*).

More rigorously: to prove (*f*(*a*_{n})) *f*(*p*), given > 0, use continuity at *p* *X* to find > 0 such that if *d*_{X}(*p*, *x*) < then *d*_{Y}(*f*(*p*),*f*(*x*)) < .

Then use convergence in *X* to find *N* **N** such that if *n* > *N* we have *d*_{X}(*x*_{n},*p*) < . For this *N* we have *d*_{Y}(*f*(*x*_{n}), *f*(*p*)) < and so we have convergence in *Y*.

In fact, as in the **R** case, there is a converse to this theorem.

**Converse**

*If all convergent sequences are mapped to convergent sequences then the function is continuous.*

More exactly: If (*x*_{n}) *x* (*f*(*x*_{n})) *f*(*p*) then *f* is continuous at *p*.

**Proof**

Suppose that *f* were not continuous at *p*. Then for some > 0 we cannot find any choice of to satisfy the continuity condition.

In particular, = 1 will not work. Hence for some point *x*_{1} we have *d*_{X}(*x*_{1} , *p*) < 1 but *d*_{Y}(*f*(*x*_{1}), *f*(*p*)) .

Similarly = ^{1}/_{2} will not work and so for some point *x*_{2} we have *d*_{X}(*x*_{2} , *p*) < ^{1}/_{2} but *d*_{Y}(*f*(*x*_{2}), *f*(*p*)) , ...

Continue like this to get a sequence (*x*_{1} , *x*_{2} , *x*_{3} , ...) with *d*_{X}(*x*_{n} , *p*) < ^{1}/_{n} but *d*_{Y}(*f*(*x*_{n}), *f*(*p*)) for each *n*. But since (*x*_{n}) has been constructed so that (*x*_{n}) *p* this contradicts the condition given in the theorem.

**Remark**

Since "nice behaviour on convergent sequences" is a necessary and sufficient condition for continuity, this can be used as the *definition* of a continuous function.

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