## 6.2. Continuous Functions | IRA |

However, if we want to deal with more complicated functions, we need mathematical concepts that we can manipulate.

Example 6.2.1:

This, like many

Definition 6.2.2: ContinuityA function is continuous at a pointcin its domainDif: given any> 0there exists a> 0such that: ifxDand| x - c | <then| f(x) - f(c) | <A function is

continuous in its domain Dif it is continuous at every point of its domain.

Continuous functions are precisely those groups of functions that preserve limits, as the next proposition indicates:

Again, as with limits, this proposition gives us two equivalent mathematical conditions for a function to be continuous, and either one can be used in a particular situation.

Proposition 6.2.3: Continuity preserves LimitsIf fis continuous at a pointcin the domainD, and{ xis a sequence of points in_{n}}Dconverging toc, thenf(x) = f(c).If

f(x) = f(c)for every sequence{ xof points in_{n}}Dconverging toc, thenfis continuous at the pointc.

Continuous functions can be added, multiplied, divided, and composed with one another and yield again continuous functions.

Example 6.2.4:

- Which of the following two functions is continuous:

- If
f(x) = 5x - 6, prove thatfis continuous in its domain.- If
f(x) = 1ifxis rational andf(x) = 0ifxis irrational, prove thatxis not continuous at any point of its domain.- If
f(x) = xifxis rational andf(x) = 0ifxis irrational, prove thatfis continuous at 0.- If
f(x)is continuous in a domainD, and{ xis a Cauchy sequence in_{n}}D, is the sequence{ f ( xalso Cauchy ?_{n}) }

While this proposition seems not very important, it can be used to quickly prove the following:

Proposition 6.2.5: Algebra with Continuous Functions

- The identity function
f(x) = xis continuous in its domain.- If
f(x)andg(x)are both continuous atx = c, so isf(x) + g(x)atx = c.- If
f(x)andg(x)are both continuous atx = c, so isf(x) * g(x)atx = c.- If
f(x)andg(x)are both continuous atx = c, andg(x) # 0, thenf(x) / g(x)is continuous atx = c.- If
f(x)is continuous atx = c, andg(x)is continuous atx = f(c), then the compositiong(f(x))is continuous atx = c.

Continuity is defined at a single point, and the epsilon and delta appearing in the definition may be different from one point of continuity to another one. There is, however, another kind of continuity that works for all points of domain at the same time.

Examples 6.2.6:

Take a look at this Java applet illustrating uniform continuity.

Definition 6.2.7: Uniform ContinuityA function fwith domainDis calleduniformly continuouson the domainDif for any> 0there exists a> 0such that: ifs,tDand| s - t | <then| f(s) - f(t) | <. Click here for a graphical explanation.

While this definition looks very similar to the original definition
of continuity, it is in fact not the same: a function can be continuous,
but not uniformly continuous. The difference is that the delta
in the definition of uniform continuity depends only on epsilon,
whereas in the definition of simply continuity delta depends on
epsilon as well as on the particular point *c* in question.

The next theorem illustrates the connection between continuity and uniform continuity, and gives an easy condition for a continuous function to be uniformly continuous.

Example 6.2.8:

- The function
f(x) = 1 / xis continuous on (0, 1). Is it uniformly continuous there ?- The function
f(x) = xis continuous on [0, 1]. Is it uniformly continuous there ?^{2}- The function
f(x) = xis continuous on^{2}[0, ). Is it uniformly continuous there ?- If
f(x)is uniformly continuous inR, and{ xis a Cauchy sequence, is the sequence_{n}}{ f ( xalso Cauchy ?_{n}) }

Theorem 6.2.9: Continuity and Uniform ContinuityIf fis uniformly continuous in a domainD, thenfis continuous inD.If

fis continuous on a compact domainD, thenfis uniformly continuous inD.

Next, we will look at functions that are not continuous.