Problem
Use the epsilon/delta definition of continuity to show that $f(x) = 2x+1$ is continuous at $x=2$.
My work
The book shows an example using a similar, linear function, and while I can certainly do the same steps, I have a hard time groking it.
Let $h = x-2$, which gives $x = h+2$.
The allowed error is $|f(x) - f(2)| = |2h| = 2|h| < \varepsilon$ after skipping some steps, and substituting $x = h+2$.
Now comes the part I'm struggling to understand.
To satisfy $2|h| < \varepsilon$, I just need an $h \ \ \mid \ \ |h| < \frac{\varepsilon}2$. Done. Can do that for any $\varepsilon > 0$.
My actual question
What exactly is $\delta$ here? I can't fully understand why $h$ was introduced.
As far as I can tell, $|h| = \delta$, but why the extra variable?