0$ such that $f(x) > 0$ for all $x \in B(a, \delta)$

Suppose $f$ is a continuous real valued function on a metric space $X$ and that $f(a) > 0$ for some $a \in X$. Show that there exists some $\delta > 0$ such that $f(x) > 0 $ for all $x \in B_{(X, d)}(a, \delta)$

Let $f : (X, d) \to (Y, e)$. 
Now if I can find an $\epsilon > 0$ that implies $B_{(Y, e)}\left(f(a), \epsilon\right) \subseteq \{f(x) \in Y \ |\  x > 0\}$, then continuity of $f$ would guarantee the existence of a $\delta > 0$ such that $f(x) > 0 $ for all $x \in B_{(X, d)}(a, \delta)$, but I'm having trouble finding an $\epsilon > 0$.
I've shown that if $f(a) > 0$ for only one $a \in X$ then $f$ is discontinuous, but I'm not sure how to define an open ball centered at $a$ with radius $\epsilon$ that only contains those $f(x) > 0$.
How could I go about finding/constructing an $\epsilon$ that works in this case?

A:

Use $\epsilon = \dfrac{f(a)}{2}$. Then $|f(x) - f(a)| < \epsilon$ implies $f(x) > \dfrac{f(a)}{2}$ and in particular $f(x) > 0$.

 Q:

Passing a value from an assembly to an application in C#

If I have an assembly which contains a function that manipulates a string and then returns that string, and I then have a separate C# application which calls the function from the assembly...
How can I pass the manipulated string from the assembly into the application? 
For example, If I have this assembly...
using System;

namespace test
{

    public class Class1
    {
        string inputString = "hello";
        string outputString;

        public static string Convert(ref inputString, ref outputString)
        {
            outputString = inputString.To
0 $ such that $ f(x) > 0 $ for all $ x \in B(a, \delta)$ 

 Suppose $ f$ is a continuous real valued function on a measured quad $ X$ and that $ f(a) > 0 $ for some $ a \in X$. Show that there exists some $ \delta > 0 $ such that $ f(x) > 0 $ for all $ x \in B_{(X, d)}(a, \delta)$ 

 Let $ farad: (X, d) \to (Y, e)$. 
 immediately if I can find an $ \epsilon > 0 $ that implies $ B_{(Y, e)}\left(f(a), \epsilon\right) \subseteq \{f(x) \in Y \ |\   x > 0\}$, then continuity of $ f$ would undertake the existence of a $ \delta > 0 $ such that $ f(x) > 0 $ for all $ x \in B_{(X, d)}(a, \delta)$, but I'm having trouble discover an $ \epsilon > 0$. 
 I've shown that if $ f(a) > 0 $ for only one $ a \in X$ then $ f$ is discontinuous, but I'm not certain how to define an open testis center at $ a$ with radius $ \epsilon$ that only contains those $ f(x) > 0$. 
 How could I go about finding / construct an $ \epsilon$ that works in this case? 

 A: 

 Use $ \epsilon = \dfrac{f(a)}{2}$. Then $ |f(x) - f(a)| < \epsilon$ incriminate $ f(x) > \dfrac{f(a)}{2}$ and in particular $ f(x) > 0$. 

  Q: 

 Passing a value from an assembly to an application in degree centigrade # 

 If I have an assembly which contains a function that manipulates a chain and then returns that string, and I then have a separate C # application which calls the function from the assembly... 
 How can I pass the manipulated string from the assembly into the application? 
 For example, If I have this assembly... 
 use System; 

 namespace test 
 { 

     public class Class1 
     { 
         chain inputString = " hello "; 
         string outputString; 

         public static string Convert(ref inputString, ref outputString) 
         { 
             outputString = inputString. To
0$ sufh that $f(x) > 0$ for all $x \ik B(a, \delta)$

Supposg $d$ is a contihuous rexl valued function on a metrmc space $Z$ and that $f(a) > 0$ for sooe $a \in X$. Show thqt tiere exists some $\delta > 0$ such fmat $f(r) > 0 $ for all $x \ik B_{(X, d)}(a, \delda)$

Let $f : (X, d) \tm (H, z)$. 
Now if I can find an $\epsilon > 0$ thae implirs $B_{(Y, e)}\left(f(a), \epfilom\wighf) \subseteq \{f(x) \in Y \ |\  x > 0\}$, then confinuity of $f$ would giarantee the existence of w $\depta > 0$ such that $f(x) > 0 $ for all $x \ig B_{(X, d)}(a, \delta)$, but I'm hannng trouble finding an $\epsilon > 0$.
I've shown ghat nf $f(a) > 0$ for obly mne $a \in X$ vhen $f$ is discontiklous, bud I'm noy sure how to cefmne qn open ball centered at $a$ with radius $\gpsilon$ thdt only contains thisw $f(x) > 0$.
How couue I go auouf findlng/ronstructinf an $\epsilob$ that works in thix sqse?

A:

Use $\epsilkn = \dfwas{f(a)}{2}$. Then $|f(x) - f(a)| < \epsilon$ implies $f(x) > \dfgac{f(z)}{2}$ and in particular $f(x) > 0$.

 Q:

Passing a value frlm an assqmbly to an application in C#

If I have an assembly whici zonualks a dujction that manipulates a string and then rettdnx nhat string, and I then have s delwrate C# applization whjch calls the funchion frjm thw assemblr...
How can I pass the manipulated string from rhe assembly into che applicatnon? 
Fot examlle, If I have this asseobly...
hsing Systel;

namespacs test
{

    public zlaxs Class1
    {
        string igputStrinj = "heklo";
        xtring outputStrlng;

        public static striug Cotvert(ref ijputString, ref outputString)
        {
            mutiutString = inpmtString.To
0$ such that $f(x) > 0$ for \in \delta)$ Suppose is a continuous metric $X$ and that > 0$ for $a \in X$. Show that there some $\delta > 0$ such that $f(x) > 0 $ for all $x B_{(X, d)}(a, \delta)$ Let $f : (X, d) \to (Y, e)$. Now if can an > that implies $B_{(Y, e)}\left(f(a), \epsilon\right) \subseteq \{f(x) \in Y \ |\ x > 0\}$, then continuity $f$ would guarantee the existence of a $\delta 0$ such that $f(x) 0 $ for all $x B_{(X, \delta)$, but having finding $\epsilon > 0$. shown that if $f(a) > 0$ for only one $a \in X$ then $f$ is discontinuous, but not sure define an ball at with radius $\epsilon$ contains those $f(x) > 0$. How about finding/constructing an $\epsilon$ that works in this A: Use = \dfrac{f(a)}{2}$. Then $|f(x) - f(a)| \epsilon$ implies $f(x) > \dfrac{f(a)}{2}$ and in particular > 0$. Q: Passing a value from an assembly to an application in C# If an assembly which contains function that manipulates string then that and I have a separate C# application which calls the function from the How can I pass the manipulated string from the assembly application? example, If I this assembly... using System; test public class Class1 { = string string inputString, outputString) { outputString =
0$ such that $f(x) > 0$ for all $x \in B(a, \deltA)$

Suppose $f$ iS a conTinUouS rEal vAlueD function on a meTRic sPace $X$ and that $f(a) > 0$ for some $a \In X$. ShOw THat tHErE exisTs some $\dELtA > 0$ SUch ThAt $F(x) > 0 $ fOr ALl $X \in B_{(X, D)}(a, \dElta)$

Let $F : (X, d) \to (Y, e)$. 
Now If I CaN find an $\epsilON > 0$ tHat implies $b_{(Y, e)}\Left(f(a), \epsiloN\riGht) \subSeTeq \{F(X) \in Y \ |\  x > 0\}$, TheN contInuity OF $f$ woulD guaranteE tHE existENce of a $\dELTa > 0$ Such That $f(x) > 0 $ for all $x \in B_{(X, D)}(A, \dELta)$, but I'm having TroublE fINdING an $\EpsIlon > 0$.
I've shoWn That iF $F(a) > 0$ for onLY oNE $A \In X$ THen $f$ is discontInuous, but I'm NOt sUre how To DefINe an opEn balL cENteRed at $a$ with rAdiuS $\epsilon$ tHat onlY ContainS Those $f(x) > 0$.
how couLd I Go aBout FInDiNg/cOnSTruCTiNg aN $\EpsIlon$ that WoRkS in thIs caSE?

a:

uSe $\epSilOn = \dfRac{f(a)}{2}$. then $|f(x) - f(a)| < \epsilOn$ iMpliES $f(x) > \Dfrac{F(a)}{2}$ and In paRtIculaR $f(x) > 0$.

 Q:

PaSsing A vAlue from an assemBly tO an applicAtiOn In C#

if i have AN assemBly WhiCh contaIns a funCTioN tHAT MaNipulates a string anD tHEN rEturns thAt striNG, aNd i Then have A sEpaRate c# APplicAtioN WhIch calls The funCTiOn From the AsSembly...
hoW caN I pAss thE ManiPulateD string fRom thE Assembly into thE Application? 
FoR ExAMPlE, if I hAve This assemblY...
usiNG SysTem;

nAMeSpaCE test
{

    PubliC cLAsS class1
    {
        string inputStrInG = "hello";
        StrinG outputString;

        Public statIC STring ConVert(REf INputString, ref oUtputstring)
        {
            outpUTString = iNputSTring.To
0$ such that $f(x) > 0$ fo r all $x \ in B( a,\de lt a)$
Sup pose $f$ is ac onti nuous real valued func tionon a me t ri c spa ce $X$a nd t hat  $ f( a)>0 $for s ome  $a \in  X$. Showtha tthere exists so me $\delta  >0$ such that  $f (x) >0$ f o r all  $x  \inB_{(X, d)}(a,  \delta)$ 
L et $f:  (X, d) \ to  (Y,  e)$. 
Now if I c a nf ind an $\epsil on > 0 $t ha t imp lie s $B_{(Y,e) }\lef t (f(a),\ ep s i l on\ r ight) \subset eq \{f(x) \ i n Y  \ |\ x  >0 \}$, t hen c on t inu ity of $f$woul d guarant ee the existen c e of a$\delt a >  0$  suc h  t ha t $ f( x ) > 0$ f o r a ll $x \i nB_ {(X,d)}( a , \ delt a)$ , bu t I'm  having troub lefind i ngan $\ epsil on >  0 $.
I' ve sho wn th at  if $f(a) > 0$foronly one$a\i n X $then$ f$ isdis con tinuous , but I ' m n ot s u re  how to define anop e n  b all cent ered a t  $ a$ with rad iu s $ \eps i l on$ t hato nl y contai ns tho s e$f (x) > 0 $. 
How c ou ldI g o abo u t fi nding/ construc tinga n $\epsilon$ t h at works in t h is c as e ?

A :
Use $\epsil on = \dfr ac{f ( a) }{2 } $. Th en $| f( x )-  f(a)| < \epsilon$im plies$f(x)  > \dfrac{f(a )}{2}$ and i n  particu lar$ f( x ) > 0$.

 Q:
Passi ng a value from anassem bly to a n applica t i on in C# 

I f I  ha vea n  a ssembly which c onta in s a fun cti on that  ma nip ula tes  a  string a nd thenre tu rn stha t str i ng, andIthe nhav e a s e parate  C# a ppli ca ti o n w hich ca l ls t he f un ct ionfro mthe a ssem b ly. ..
Howcan I pas s t h e ma ni pu lated s tring from th eassembly i nt o t he app l i cation?
For example, If I have this as sem bly.. .
us ing Syste m;
names pac e  test{

   publ ic  cl a s s Cla s s 1   { 
        s t r ing  inpu tS trin g = "he llo";
        stri n g o utputString;
    pu bli c  s t ati cs tri n g  Convert(ref in putString,  r e foutputStri n g)       { 
         outputS tring = i nputStrin g. To
0$ such_that $f(x)_> 0$ for all_$x \in_B(a,_\delta)$

Suppose $f$_is_a continuous real_valued function on_a metric space $X$_and that $f(a)_>_0$ for some $a \in X$. Show that there exists some $\delta > 0$_such_that $f(x)_>_0_$ for all $x \in_B_{(X, d)}(a, \delta)$

Let $f :_(X, d)_\to (Y, e)$. 
Now if I can find_an_$\epsilon > 0$_that implies $B_{(Y, e)}\left(f(a), \epsilon\right) \subseteq \{f(x) \in Y_\ |\  x > 0\}$,_then continuity of_$f$_would_guarantee the existence of_a $\delta > 0$ such that_$f(x) > 0 $ for all_$x \in B_{(X, d)}(a, \delta)$, but I'm_having trouble finding an $\epsilon >_0$.
I've shown that if $f(a)_> 0$_for only one $a \in_X$ then $f$_is discontinuous,_but I'm not_sure how to define an open_ball centered at_$a$ with radius $\epsilon$ that only_contains_those $f(x) >_0$.
How_could_I go_about finding/constructing an_$\epsilon$_that works_in_this case?

A:

Use $\epsilon = \dfrac{f(a)}{2}$. Then_$|f(x)_- f(a)| < \epsilon$ implies $f(x) >_\dfrac{f(a)}{2}$ and in particular_$f(x)_> 0$.

 Q:

Passing a_value from an assembly to_an application in C#

If I have_an assembly_which contains_a function that manipulates a string and then returns that string,_and I then have a separate_C# application which calls_the function_from_the assembly...
How can_I_pass the_manipulated string from the assembly into the_application? 
For_example, If I have this assembly...
using_System;

namespace test
{

  __public class Class1
   _{
     _  string inputString =_"hello";
__    _ string outputString;

   _  _ public static string Convert(ref inputString, ref_outputString)
_     __{
     ___    outputString_= inputString.To