Can you find f,g st sup{|f-g|}=0 but f is not equal to g

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Homework Help Overview

The discussion revolves around the existence of functions f and g such that the supremum of the absolute difference |f(x) - g(x)| is equal to zero, while also asserting that f is not equal to g. This involves concepts from real analysis, particularly dealing with the properties of functions and their limits.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the meaning of the supremum and its implications, questioning the interpretation of the statement that f is not equal to g. There is an attempt to clarify the definition of supremum and its distinction from maximum.

Discussion Status

The discussion is ongoing, with participants raising questions about the implications of the definitions involved. Some guidance has been offered regarding the interpretation of supremum and the conditions under which f and g can be considered equal or not.

Contextual Notes

There is a noted ambiguity in the problem statement regarding the conditions under which f and g are defined, particularly concerning the assertion that f is not equal to g while maintaining a supremum of zero.

happybear
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Homework Statement


For all real number x, can you find a function f and g such that
sup|f(x)-g(x)|=0

Homework Equations





The Attempt at a Solution

 
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happybear said:

Homework Statement


For all real number x, can you find a function f and g such that
sup|f(x)-g(x)|=0

Homework Equations





The Attempt at a Solution

What have you done on this problem yourself? In particular, what does sup |f(x)- g(x)| mean? And if you are going to put "f is not equal to g" please put it in the statement of the problem!
 
HallsofIvy said:
What have you done on this problem yourself? In particular, what does sup |f(x)- g(x)| mean? And if you are going to put "f is not equal to g" please put it in the statement of the problem!


sup{|f-g|} means that the maximum of the value. I try to find a function f approaching g, but this eem not to be the case
 
happybear said:
sup{|f-g|} means that the maximum of the value. I try to find a function f approaching g, but this eem not to be the case
Strictly speaking, sup does NOT mean 'maximum', it means "least upper bound" which may or may not be a maximum. What do you mean "f approaching g"? As x approaches what value? This has to be true over all x.

Suppose f were NOT equal to g. Then there must exist some x such that [itex]f(x)\ne g(x)[/itex]. Let M= |f(x)- g(x)| for that x. Now, what can you say about sup|f(x)- g(x)|?
 
so does that mean that no matter whether X is compact or not, there is no such a function?
 

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