It is straightforward to show the continuity of the polynomials, and so we hate that silly [citation needed] on Polynomial » Polynomial functions.

Weierstrass? Great choice! Great choice! Let's go!

Unfolding Weierstrass's continuity, one observes that any monomial,

is continuous, and then use the standard result that the sum of two continuous functions on some domain, in this case , is also continuous on that domain. Iteratively applying the result permits the conclusion that the sum of a finite number of monomials,

is continuous, i.e. that any polynomial is continuous.

Doc's not fucking having it. He says the fucking result for the case x0 = 0 is so obvious (δ = | ε |1/n) it's not worth worrying about, so he says to get your shit in gear and forget about x0 = 0 already:



Well, we're talking monomials, so we have:


So let ε > 0 be given. We need to show δ > 0 such that


First we note that,


Next we use the binomial expansion:

So we get,


There are n terms in this sum, so we can say that

To guarantee the above, we just pick


OK, so we've just shown any monomial is continuous. Now, as we said at the outset, we just use the fact that, loosely speaking, f + g is continuous if f and g are cts. And we're done. LudicrousTripe (talk) 01:30, 12 November 2013 (UTC)