The following problem seems very simple, and indeed I'm sure the answer isn't hard at all. I know how to prove the limits of polynomial functions, using the epsilon-delta definition of a limit, by factoring certain parts, using certain theorems about sums and products of limits, and by finding upper bounds for certain expressions. From my understanding, the basic "approach" to finding the following:

$\displaystyle \lim_{x \to a} f(x)$

is to find a factor that puts the function in the following form:

$\displaystyle f(x) = (x-a)g(x) + b$

for some numbers $\displaystyle a$ and $\displaystyle b$, and some function $\displaystyle g(x)$. My problem is that I don't know how to go about doing so for the following problem:

Prove the following using the epsilon-delta definition of a limit:

$\displaystyle \lim_{h \to 0} f(a+h) = f(a)$

Now, obviously this limit is "intuitively" true but I can't seem to figure out the exact definition of delta in the proof. I've done some scratch work, I know that generally this forum encourages that posters display any work they've done themselves so far, so I'll put that below. Granted, it may contain errors:

I started with this:

$\displaystyle |h-0|=|h| < \delta$

and made the stipulation (with no particular motivation other then finding an upper bound on something):

$\displaystyle |h|< 1$

therefore:

$\displaystyle -1<h<1$

the next line is something that I'm not certain is "always" true. In other words, I'm not certain if it follows from the above line:

$\displaystyle f(a-1)<f(a+h)<f(a+1)$

now, I know I need to somehow get to the point of having this statement:

$\displaystyle |f(a+h) - f(a)|< \epsilon$

but I'm not sure how to proceed correctly. Do I subtract $\displaystyle f(a)$ from all parts of the inequality (the one three lines above) above? Or should I start by working firstly with $\displaystyle |f(a+h) - f(a)| < \epsilon \;\;$?

Any guidence would be much appreciated.