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

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SUMMARY

The discussion centers on the mathematical concept of finding functions f and g such that the supremum of the absolute difference |f(x) - g(x)| equals zero, while f is not equal to g. Participants clarify that "sup" refers to the least upper bound, not necessarily the maximum, and emphasize that if f is not equal to g, there must exist at least one x where f(x) ≠ g(x), contradicting the condition that sup|f(x) - g(x)| = 0. Consequently, it is concluded that no such functions can exist under these conditions.

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  • Knowledge of the definition of absolute value and its implications.
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  • Study the concept of supremum in real analysis, focusing on its properties and applications.
  • Explore examples of functions to understand conditions under which |f(x) - g(x)| can equal zero.
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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 f(x)\ne g(x). 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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