Finding Discontinuous Composite Function f o g at x = 0

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SUMMARY

The discussion focuses on finding continuous functions f and g at x = 0 such that their composite function f(g(x)) is discontinuous at x = 0. A key insight is that the discontinuity arises when the composite function results in a scenario where 0 is in the denominator, rendering it undefined. The participants confirm that while f and g can be continuous individually, their composition can still lead to discontinuity, specifically at x = 0.

PREREQUISITES
  • Understanding of composite functions in calculus
  • Knowledge of continuity and discontinuity in functions
  • Familiarity with limits and undefined expressions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Research examples of continuous functions that yield discontinuous composites
  • Study the implications of the limit definition of continuity
  • Explore cases where the denominator of a function approaches zero
  • Investigate the behavior of piecewise functions and their composites
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in understanding the nuances of function composition and continuity.

Jan Hill
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Homework Statement


I need to find functions f and g both continuouis at x = 0 for which the composite f at g is discontinuous at x = 0



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The Attempt at a Solution

I thnk it is a matter of looking for a composite function that results in 0 being in the denominator so that it's undefined. Is that the only way to get this composite to be discontinuous?
 
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and can it be that f\circ g is discontinuous at x = 0 if f and g are both continuous at x = 0 despite the fact that we know that g\circ f cannot be discontinuous at x = 0 if f and g are continuous at x= 0
 

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