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hanin

hanin

New Member
السلام عليكم
ارجو من الاخوة الكرام مساعدتي باثبات مايلي:
show that if f is continuous function , and z(f) is the set of all zeros of f and A is the set of all limit points of z(f) then A is subset of z(f) . thanx
 


let f : X ---Y be a cts function, and let Z(f) be the set of all zeros of f

let Z'(f) be the set of of all limit point of Z(f) .... want Z'(f) subset of Z(f)??????try


suppose not, therefore there is x in Z'(f) which is not in Z(f), so that

let U(x) be an open set containing x subset of X, then

U(x) intersecion Z(f)\{x} not equal to empty or phi

since f is cts therefore

f(U(x) intersection Z(f)\{x} ) not equal to f(phi) but since f(phi) = 0, therefore


f(U(x) intersection Z(f)\{x}) not equal to 0

which is contradicts our assumptions
 


thank u {{alomari}}} I tried to prove it and then i reached the same result
thanx a lot for concern
 



let x in A, then x is a limit point of Z(f) , so there is a sequence (x_{n}) in Z(f) such that (x_{n}) goes to x as n go to infinity

Now by continuity of f, we have f (x_{n}) goes to to f(x) , but (x_{n}) in Z(f), so f (x_{n}) is a sequence of zeros and so f(x) must be zero
Then x in Z(f), therefore A is a subset of Z(f) . ok)


نسأل الله أن يكون نافعاً بحول الله​
 


شكرا ابو رشاد على البرهان الكافي الوافي. بصراحة ما خطر لي استخدم ال sequence بس هيك الحل مختصر وصحيح
شكرا مرة تانية
 
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