Def let d and g ke2 metrics defined on E

c We call g is stthan d or d is neaterthan if if I 0 Sit

dexy E CgCxy F x yc I

They are equivalent if g is stronger andweaker than d ie 7 Cs Cz o set

dex y E C pay E C duy F X yc I

2

or c dex y E gex y E a daisy7 MycE

Prof d If f is stronger than d then

Haig converges in CE g vinplies

Hag converges in E dand hence

the same bin it2 If f isequivalent to d then Krisconverges in g if and only if I ingconoerges in C E d

3 equivalent ofmetrics defined aboveb an equivalent relation

Pf Easy ex

Cf Oa IR d y ElXi Yi

dux y El Xi yayJkmqxlxiyildcocx.gs

Checkc's day sin day Ern ddY9

cis's diary E n day End c b

Therefore di da do are equivalent metrics

on IR

ef E Cta b d Cf g S If 91

discs g megaby If 91

Then clearly

diff S E b a duct S3 Ffg C Cta bJ

dos is stronger than dis

However it is airpossible to find C o set

d Csg E Cd Cf g V fgtc a.br

If Before fa on Fa b 10,1

fa x e fMX th X CEoat

h

To X Ent B

o 1

Then d Cfu D f lfat In 70 as n too

dasCfn 0 EggHonkyn

i n d Cfn O E C d Hn o Ean th

which is utopasibled is not stronger than da

Therefore di ados are not equivalent

Def let fi CE d CF g be a mapping between

2 metric spaces and X E E We call f

is continuity at x if

f Ha fix in CT fwhenever Xu x in Ed

It a continuous on a set E CE f it is continuous

at every pointof E

Proper let filk d T g be a mapping between 2metric spaces and XoC E Then

f 5 continuous at Xo

t E so I 870 such that

I f Cfcs fexo s E F Xwith dasXo Ef

Pf Ex

Propes i let f tx d CFge

g e CFp Z m

are mappings between metric spaces

9 If f is continuous atx e gis centaicas atfag

then go f I d Z mis contineous at x

If f is Is in E and g is ctsin'T

then gof is is in E

Pf Easy

Eg let fE d be a metricspace ACE AFORefire PA I R by

Saks imf dayx

YEAdistance from X to the subset A

Clad I faks pacy I E da y tx YEE

PIifclain For fixed ye E

Bydefn of PATH

HE 0 I 2C A set PA t E de y

Hence fax s de X E du y digitday Pacy E

faced facy s dcx.gsE

Interchanging therolesof X y

fab PAIDs dasyste

Therefore 1pA paly I s d ItE

Sina E 0 is arbitrary IPAID falystEdgy

By claim dan x o fauns Paks

is dR b cts

In fact Pa is Lipschitzcontinuous

The example shows that there are many cts

functions on a metric space

Notation Usually we use the following notations

das F tuff dayy y EF

dCE F infldcyy X EE DEF

for subsets Ee F