Are all connected and locally integral affine schemes globally integral? In these notes, on p. 2 Section 4, Kedlaya claims an affine scheme is integral if and only if it is connected and every local ring is an integral domain. But elsewhere I have seen that this requires a Noetherian condition on the affine scheme, e.g. problem 19 here, problem 56D in this. 
What is wrong with Kedlaya's proof? Or is the result actually true without assuming Noetherian? 

The error is in the exercise referred to in the first paragraph: it is not necessarily true that the set of $x$ such that $f$ is nonzero in $O_{X,x}$ is open.  In fact, this can fail even if $A$ is Noetherian.  For instance, let $k$ be a field and take $A=k[x,y]/(xy)$, so $X$ is the union of two lines intersecting at a point.  Then the set where $x$ is locally nonzero is one of the lines, which is not open since it contains no neighborhood of the intersection points.
For a counterexample to the result without the Noetherian hypothesis, see https://stacks.math.columbia.edu/tag/0568.
