Prove that f is continuous on (a, b), with a property given?

  • Thread starter LilTaru
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  • #1
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Homework Statement



Suppose the function f has the property that |f(x) - f(t)| <= |x - t| for each pair of points x,t in the interval (a, b). Prove that f is continuous on (a, b).


Homework Equations



I know a function is continuous if lim x-->c f(x) = f(c)


The Attempt at a Solution



I have no idea how to even start this question. I know a function is continuous on an open interval if it is continuous for all interior points, but how do I even begin to show that? Please help?!
 

Answers and Replies

  • #2
HallsofIvy
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Yes, you need to prove that [itex]\displaytype\lim_{x\to x_0} f(x)= f(x_0)[/itex] for any [itex]x_0[/itex] in (a, b). That, from the basic definition of limit, is the same as showing that "Given [itex]\epsilon> 0[/itex], there exist [itex]\delta> 0[/itex] such that if [itex]|x- x_0|< \delta[/itex] then [itex]|f(x)- f(x_0)|< \epsilon[/itex]".

But you are given that [itex]|f(x)- f(x_0)|< |x- x_0|[/itex]! Taking [itex]\delta= \epsilon[/itex] works.
 
  • #3
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Two questions:

1) So I can replace t with x0? As in instead of |f(x) - f(t)| like the question states... use |f(x) - f(x0|?

2) Why is it en? I thought it is just supposed to be < e?
 
  • #4
Char. Limit
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That's epsilon followed by closing quotation marks, not [itex]e^n[/itex]
 
  • #5
HallsofIvy
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Two questions:

1) So I can replace t with x0? As in instead of |f(x) - f(t)| like the question states... use |f(x) - f(x0|?
You said "|f(x) - f(t)| <= |x - t| for each pair of points x,t in the interval (a, b)" and [itex]x_0[/itex] is a point in (a, b)

2) Why is it en? I thought it is just supposed to be < e?
Thanks, Char. Limit, yes, that is not an 'n' it is just an end of the " ".
 
  • #6
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Oh, okay! That clears it up a lot! It works and I solved it! Thank you both for the very quick responses and help! Much appreciated!
 

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