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

[LilTaru]

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?!

 

[HallsofIvy]

Yes, you need to prove that \displaytype\lim_{x\to x_0} f(x)= f(x_0) for any x_0 in (a, b). That, from the basic definition of limit, is the same as showing that "Given \epsilon&gt; 0, there exist \delta&gt; 0 such that if |x- x_0|&lt; \delta then |f(x)- f(x_0)|&lt; \epsilon".

But you are given that |f(x)- f(x_0)|&lt; |x- x_0|! Taking \delta= \epsilon works.

 

[LilTaru]

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?

 

[Char. Limit]

That's epsilon followed by closing quotation marks, not e^n

 

[HallsofIvy]

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 x_0 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 " ".

 

[LilTaru]

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!