the Sobolev embedding theorem
We bought our freedom by working with weak derivatives, but at a price: a function in a Sobolev space might not even be continuous, let alone classically differentiable. The Sobolev embedding theorem is the refund. It says that if you have enough weak derivatives in an L^p sense, then for free you also get genuine smoothness or boundedness — you can exchange integrability of high derivatives for honest pointwise regularity.
There are two flavours. The first trades derivatives for better integrability: a function in W^{1,p} on a domain in R^n with p below n actually lies in L^q for a larger exponent q = np/(n - p) (the Sobolev conjugate) — so one weak derivative buys you membership in a strictly better Lebesgue space. The second, more dramatic flavour kicks in once you have enough derivatives relative to the dimension: if k - n/p > 0, then W^{k,p} embeds into the space of bounded continuous functions (indeed Holder-continuous ones) — the function is, after correcting it on a measure-zero set, literally continuous. The rule of thumb is the comparison of k (derivatives) against n/p (dimensional cost): each derivative is worth n/p 'units' of smoothness, and once the derivatives outweigh the dimension you cross from mere integrability into continuity.
This is the bridge between the soft, functional-analytic world (where we proved a weak solution exists) and the hard, classical world (where we want an actual smooth solution). The regularity-bootstrap argument runs exactly here: show a weak solution has many weak derivatives, then invoke Sobolev embedding to conclude it is continuous or C^1 or smoother, until it qualifies as a classical solution. The catch worth remembering: in high dimensions n you need correspondingly more derivatives to force continuity — in dimension 1, H^1 functions are already continuous, but in dimension 2 an H^1 function can be unbounded (like log log of 1/r near a point), so 'one derivative' is not enough once n is 2 or more.
In dimension n = 1, the embedding says H^1(a, b) sits inside the continuous functions: every function with one square-integrable weak derivative on an interval is (a representative is) continuous. Concretely, u(x) - u(y) = integral from y to x of u' dt, and by Cauchy-Schwarz |u(x) - u(y)| is at most (square root of |x - y|) times ||u'||_{L^2}. That is Holder continuity with exponent 1/2 — falling straight out of one weak derivative.
In one dimension, a single square-integrable derivative already forces Holder-1/2 continuity.
The exponents are sharp and dimension-dependent: the borderline case k = n/p is genuinely subtle — W^{1,n} just BARELY fails to embed into the bounded continuous functions (the log-log counterexample), so you cannot assume 'enough derivatives' without checking k - n/p > 0 strictly. The theorem also needs the domain to be reasonable (bounded with a Lipschitz boundary, say); on wild domains the embeddings can fail.