Proving Continuity of g(x) at x=3: Using Real Definition and Limit Method

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SUMMARY

The discussion centers on proving the continuity of the function g(x) defined as f(x) = 2x for rational x and f(x) = x + 3 for irrational x, specifically at the point x = 3. Participants confirm that g(x) is continuous at x = 3 and suggest using the epsilon-delta definition of continuity to demonstrate this. The consensus is that x = 3 is the only point of continuity for g(x), as the behavior of the function diverges elsewhere.

PREREQUISITES
  • Understanding of the epsilon-delta definition of continuity
  • Familiarity with piecewise functions
  • Basic knowledge of limits in calculus
  • Graphing functions to analyze continuity
NEXT STEPS
  • Study the epsilon-delta definition of continuity in detail
  • Learn how to analyze piecewise functions for continuity
  • Explore limit proofs in calculus, focusing on real-valued functions
  • Practice graphing piecewise functions to identify points of continuity
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Students studying calculus, particularly those focusing on continuity and limits, as well as educators seeking to clarify the epsilon-delta approach in real analysis.

Math_Geek
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HELP! I do not understand!

Homework Statement


Define f(x)=2x, x is rational and x+3 when x is irrational. Find all points where g(x) is continuous and prove continuity at these points


Homework Equations


From analysis homework and using the real definition of continuity


The Attempt at a Solution



Graphing g(x) I can see that the only point that will be continuous will be at x=3, how would you show using the definition that this point is continuous. Also is this the only point. I have been struggling, please help a girl in distress. lol
 
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Math_Geek said:
Graphing g(x) I can see that the only point that will be continuous will be at x=3, how would you show using the definition that this point is continuous. Also is this the only point. I have been struggling, please help a girl in distress. lol

:smile: … don't worry that pretty little head … :smile:
​

Just choose an epsilon, and find a delta such that f(3 + x) - 6 < epsilon, for all x < delta, and all your worries will go away! :smile:
 
so I just solve it like a limit problem? Thank you so much, I just knew someone would help a girl in distress lol.
 

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