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 $\epsilon> 0$, there exist $\delta> 0$ such that if $|x- x_0|< \delta$ then $|f(x)- f(x_0)|< \epsilon$".

But you are given that $|f(x)- f(x_0)|< |x- x_0|$! 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!