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

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
Homework Helper
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.

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

HallsofIvy
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)