I Understanding uniform continuity....

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The discussion centers on proving that a continuous function f is uniformly continuous on the interval [a,c] given that it is uniformly continuous on [a,b] and [b,c]. A key point is the necessity of demonstrating continuity at the point b, as discontinuity could prevent uniform continuity across the combined interval. The participants explore the epsilon-delta definition of uniform continuity and the implications of continuity at b, emphasizing that continuity must hold in a neighborhood around b to ensure the function behaves uniformly across the entire interval. The conversation also highlights the importance of using two points in the proof to establish the relationship between the two intervals. Ultimately, the proof hinges on confirming that continuity at b allows for the merging of the two intervals into a uniformly continuous function on [a,c].
  • #31
Alpharup said:
Am fully convinced now. Thanks Stephen and Lavinia. g on [a, b] is continuous, so there is left hand continuity at b. Similarly h has Right hand continuity at b. We exploit this property. Once again, am I on right track?

I think you are on the right track, but things would be clearer if you present a self-contained statement of what is to be proven as well as the proof. The text you quote from Spivak refers to "statements (i) and (ii)" without showing what they are.
 
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  • #32
You are right. I should have posted it.
Here is definition of uniform continuity.
Capture1.PNG
 
  • #33
Also about (i) and (ii)
)
Capture2.PNG
 
  • #34
Capture3.PNG
 

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