Consider the function

f:R->[0,1]

defined by

f(x)=1

if

x

is rational, and

f(x)=0

if

x

is irrational. For which values of

ainR

does\

\\\\lim_(x->a)f(x)

exist? At which values of

ainR

is

f

continuous?\ Answer the same questions for the function

g:R->[0,1]

defined by

g(x)=(1)/(n)

if

x

is rational and

x=(m)/(n)

in lowest terms (with

n>0

), and\

g(x)=0

if

x

is irrational. (To avoid ambiguity, let's assume that we write

0=(0)/(1)

, so that

n=1

in this case.)

1. Consider the function f:R[0,1] defined by f(x)=1 if x is rational, and f(x)=0 if x is irrational. For which values of aR does limxaf(x) exist? At which values of aR is f continuous? 2. Answer the same questions for the function g:R[0,1] defined by g(x)=n1 if x is rational and x=nm in lowest terms (with n>0 ), and g(x)=0 if x is irrational. (To avoid ambiguity, let's assume that we write 0=10, so that n=1 in this case.)