Continuity, proving that sin(x)sin(1/x) is continuous at 0.

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SUMMARY

The function f(x) = sin(x)sin(1/x) for x ≠ 0 and f(0) = 0 is continuous at x = 0. To prove this, one must apply the definition of continuity, which requires showing that the limit of f(x) as x approaches 0 equals f(0). Despite sin(1/x) oscillating infinitely as x approaches 0, it remains bounded between -1 and 1, allowing the product sin(x)sin(1/x) to approach 0, thus confirming continuity at that point.

PREREQUISITES
  • Understanding of the definition of continuity in calculus.
  • Knowledge of the properties of the sine function, particularly sin(x) and its behavior near zero.
  • Familiarity with limits and the concept of bounded functions.
  • Basic experience with oscillatory functions and their limits.
NEXT STEPS
  • Study the definition of continuity in more depth, focusing on epsilon-delta proofs.
  • Learn about bounded functions and their implications in limit calculations.
  • Explore the behavior of oscillatory functions like sin(1/x) as x approaches 0.
  • Review examples of proving continuity for piecewise functions.
USEFUL FOR

Students studying calculus, particularly those focusing on continuity and limits, as well as educators looking for examples of continuity proofs involving oscillatory functions.

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Homework Statement



Define f(x)=sin(x)sin(1/x) if x does not =0, and 0 when x=0.

Have to prove that f(x) is continuous at 0.

Homework Equations



We can use the definition of continuity to prove this, I believe.



The Attempt at a Solution



I know from previous homework assignments that sin(x) is continuous at 0, but sin(1/x) is not. Is there a way I can use this knowledge to help me solve this problem, or do I need to start from scratch? If I need to start from scratch, do I apply the definition of continuity?
 
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Apply the definition of continuity, and think about what these functions do as they approach zero. While sin(1/x) oscillates wildly, it remains bounded. So what is the limit of your function as x->0?
 

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