First fundamental theorem, question about continuity

Hi,

I was reading the proof for the first fundamental theorem in Apostol, and one part of it says the following.

Continuity of $\displaystyle f$ at $\displaystyle x$ tells us that, if $\displaystyle \epsilon$ is given, there is a positive $\displaystyle \delta$ such that

$\displaystyle |f(t)-f(x)|<\frac{1}{2}\epsilon$

whenever

$\displaystyle x-\delta < t < x+\delta$

I do not see how $\displaystyle |f(t)-f(x)|$ must be less than $\displaystyle \frac{1}{2}\epsilon$. I do see why it has to be less than $\displaystyle \epsilon$ though..

Any hints?

Thanks.