Support of Function: Closure of Non-Zero Set

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SUMMARY

The support of a function is definitively defined as the closure of the set where the function is nonzero, rather than simply the set where the function is nonzero. This distinction is crucial as it prevents ambiguity in cases where the function oscillates, such as with sin(x), where it crosses the axis frequently. Some literature may define support differently, but the closure definition is widely accepted for clarity and consistency in mathematical analysis.

PREREQUISITES
  • Understanding of mathematical functions and their properties
  • Familiarity with the concept of closure in topology
  • Knowledge of continuity and discontinuity in functions
  • Basic understanding of oscillatory functions, such as sin(x)
NEXT STEPS
  • Research the concept of closure in topology and its implications in analysis
  • Study the properties of oscillatory functions and their behaviors
  • Examine different definitions of support in various mathematical texts
  • Learn about continuity and discontinuity in the context of function support
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Mathematicians, students of advanced calculus, and anyone studying functional analysis will benefit from this discussion on the definition and implications of function support.

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the support of a function is defined as the closure of the set where f is nonzero

my question is why not just define the support as the set where f is nonzero?

thanks
 
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yifli said:
the support of a function is defined as the closure of the set where f is nonzero

my question is why not just define the support as the set where f is nonzero?

thanks
It could get messy if the function crosses the axis a lot, like sinx.
 
yifli said:
the support of a function is defined as the closure of the set where f is nonzero

It isn't defined that way in all books. Some books define it to be the set where the function is nonzero.
 

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