> f is Riemann integrable iff it is bounded and continuous almost everywhere.
FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". I like that characterization b/c it seems more precise than "almost everywhere", but I've heard both.
I mention that because when I read the first footnote, I thought this was a mistake:
> boundedness alone ensures the subinterval infima and suprema are finite.
But it wasn't. It does, in fact, insure that infima and suprema are finite. It just does NOT ensure that it is Riemann integrable (which, of course the last paragraph in the first section mentions).
Thanks for posting. This was a fun diversion down memory lane whilst having my morning coffee.
If anyone wants a rabbit hole to go down:
Think about why the Dirichlet function [1], which is bounded -- and therefore has upper and lower sums -- is not Riemann integrable (hint: its upper and lower sums don't converge. why?)
Then, if you want to keep going down the rabbit hole, learn how you _can_ integrate it (ie: how you _can_ assign a number to the area it bounds) [2]
> FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities".
It is not: for example, the piece-wise constant function f: [0,1] -> [0,1] which starts at f(0) = 0, stays constant until suddenly f(1/2) = 1, until f(3/4) = 0, until f(7/8) = 1, etc. is Riemann integrable.
"Continuous almost everywhere" means that the set of its discontinuities has Lebesgue measure 0. Many infinite sets have Lebesgue measure 0, including all countable sets.
The indicator function of the Cantor set is Riemann integrable. Like you said, though, the Dirichlet function (which is the indicator function of the rationals) is not Riemann integrable.
The reason is because the Dirchlet function is discontinuous everywhere on [0,1], so the set of discontinuities has measure 1. The Cantor function is discontinuous only on the Cantor set.
"Almost everywhere" is precisely defined, and it is broader than that. E.g. the real numbers are almost everywhere normal, but there are uncountably many non-normal numbers between any two normal reals.
I think it's debatable whether or not it's an ad. I also think it's debatable whether or not the title of the post is sensational.
BUT
It truly warms my heart to see the level of mistrust the comments in this thread show towards (a) venture capital funding and (b) anything even resembling an ad.
Thats going to make any LLM agent change from "cool, we can deploy this anywhere" to "it only works on this one specific paid service thats overkill and more expensive for basically everyone" - its deceptive more than anything.
> I think most people would interpret “scanning your computer” as breaking out of the confines the browser and gathering information from the computer itself.
That is exactly how I interpreted it, and that is why I clicked the link. When I skimmed the article and realized that wasn't the case, I immediately thought "Ugh, clickbait" and came to the HN comments section.
> To reiterate, at no point am I saying this is good or acceptable. I think there’s a massive privacy problem in the tech industry that needs to be addressed.
100% Agree.
So, in summary: what they are doing is awful. Yes, they are collecting a ton of data about you. But, when you post with a headline that makes me think they are scouring my hard drive for data about me... and I realize that's not the case... your credibility suffers.
Also, I think the article would be better served by pointing out that LinkedIn is BY FAR not the only company doing this...
That sounds problematic and is only supported by people mindlessly agreeing to it. I know someone who got jobs at google and apple with no linkedin, and he wasn't particularly young. What do you do in the face of it? I say quit entirely. It was an easy decision because I got nothing out of it during the entire time I was on it.
After getting laid off at age 52 (2nd time, 1st time day after my 50th birthday, took an inter-company transfer), and searching for a year, applying to maybe 5-10 companies a week, I got my current job (2 years+) through a random LinkedIn button.
I'm glad to see this comment and the parent comment voted so near the top. I've had the same experience. In my experience, Kagi used to be great... then it became good... and now it's "better than Google".
"Better than Google" and the fact that I can choose websites to exclude from my search results are two features that I remain willing to pay for, however.
I'm extremely confused by these comments. Are we all using the same google? Just to make sure I wasn't crazy I just did a search on Google and 1/2 the page was a combination of google AI result and ads. Below that there were 2.5 links visible. One reddit result, and two blogspam.
The exact same search on Kagi ('best lllm for coding') nets reddit, hacker news, and some other forum results right at the top, followed by a long dense list of links to various sites (including some of the same blogspam of course), but over all the results are hugely more rich and varied and also not at all the same.
How can you possibly say that a site that gives you 50% ads and a bunch of low quality links is remotely "only a little better" than a site that gives you zero ads and a huge number of better quality links?
I'm glad to see this comment and the parent comment and the grandparent comment voted so near the top. I've had the same experience.
I honestly would love to be able to give my Kagi key to the ChatGPT or Claude clients (or more realistically, configure a proxy) just to have it be their primary tool for searches—respecting my site rankings/lists
I’m confused by this comment. The original comments talk about Kagi not living up to the hype. You say you’ve had the same experience and wish you could get LLMs to use Kagi for web searches?
Especially odd as that’s exactly what Kagi assistant already does. Maybe they’d just rather use their key than pay Kagi for LLM based search.
On that note, Kagi research is legit amazing. There have been times I’ve spent 30min searching for something without success. As a last resort I asked Kagi research and it found why I could not. More than one option even. Now intend to use almost more than normal search.
Yeah, I agree with other comments here that the traditional search offering has gotten a bit worse (I think because the whole web has gotten worse), but research surfaces great results. Forums, small blogs and websites with authoritative views on subjects I search for. Really great. Yes, it is as expensive as any other AI pro plan unfortunately, but worth it for me.
I noticed this years ago with technical books. IIRC, Manning was the first publisher that I noticed doing it. Pages so thin that I could see the text on the reverse side as I was reading it - it drove me crazy. O'Reilly started doing the same.
I had a PDF version of On Lisp (Paul Graham put it on his website for free some time after it went out of print). I used lulu.com to turn it into a printed book (1 copy for myself). I love it. The cover art isn't great (low-res image; not Lulu's fault), but the paper stock is amazing (I got to choose it!). The print quality is also great.
Lulu provides some evidence that you can run a profitable business and still offer users the ability to do _very small_ print runs (1 book). I wish they (or someone like them) could work out a deal with publishers that would let me choose the paper stock I want when I order a book online.
But, maybe there are other options...
Two quotes from the article:
> I purchase most of my books through Amazon. I don’t find the speed of delivery that valuable, but the competitive pricing (especially factoring in Prime), ease of ordering
[...]
> To add insult to injury, print-on-demand books seem to be significantly more expensive than stock equivalents
That's the classic enshittification playbook right there. Hook 'em with low prices. Once you've captured the market, lower your costs and raise prices.
Vote with your wallet. Go to a bookstore. Small and local is fun if you don't have a particular book in mind. If you do have a particular book in mind, check Barnes and Noble's website. It will tell you if it's in stock near you. If not, order it. If you go to pick it up and don't like the quality of the print/binding -- return it.
It's annoying because Lulu actually has a storefront where you can buy books they print - but only in the versions, sizes, paper, and options that the author/publisher picked: https://www.lulu.com/shop
I think we can agree that vinyl sounds different than CD, right? Is it so hard to believe that some people actually prefer the sound of music on vinyl? For such a person, that might be the only metric that matters.
But, another example: when I was growing up (dating myself here), cassette tapes were superior to CDs in the only way that mattered (to me): they didn't skip in my portable music player (walkman) when I took them running.
The sound of vinyl is a subset of the sound of CD. If you take a high quality recording of a vinyl record playback and write it to CD, it will sound identical.
Okay, sure. But if I prefer the subset of CD sound that is the same as vinyl, and my favorite band comes out with a new album... I just buy vinyl, right?
Or are you suggesting that I buy the record, a blank CD, and all of the high quality recording playback equipment I need to write it to that CD?
But my question is, how long has this been going on for? I'm in my late 40s. I remember as a kid thinking Schwinn bicycles were the "cheapo" brand. My father, however... would would be in his 90s now... remembered them as a top tier brand.
I have this theory that the reason a lot of prices on goods we've loved our whole lives don't keep up with inflation is because brand loyalty has SUCH power that it's worth it for people to buy and enshittify those brands -- so that they can sell them to us as we age at the prices we are used to paying for them.
They make newer "luxury" brands to sell to younger people who are still deciding what a "reasonable" price to pay for something is.
> So, my question is: are the people that are upset with the "ambiguity" people who neither
> (a) want to buy a license nor
> (b) be bound by the AGPLv3?
No and no. People first want to know what the correct licenses are even before deciding which licensing path (including buying a commercial license) to take. You don't just commit to buying a commercial license without first understanding your options and comparing those options. People want to know what those options are.
People are upset that a company cannot get the simple matter of open source licensing right. It's the easiest kind of licensing. But they cannot get it right. These upset people would now never want to do business with this company.
People who would have otherwise been happy to purchase a commercial license would also stay away from the company because messing up open source licensing is a red flag. Who knows what kind of mess would be present in their commercial contracts. Yes, you can hire a lawyer to sort it out but I'd much rather do business with a company where I'm confident that the company is acting in good faith even before lawyers get involved.
> If so, I have no sympathy.
Your sympathy means nothing to me when I am picking vendors for my business. When I'm picking my vendors, I'm going to rely on professional legal expertise available to me, not the sympathies of random strangers on the internet.
> On the other hand I am also a realist and I don't think that Linux will take over the Desktop, but it will certainly have its biggest growth year ever in 2026.
I _love_ Linux, but I agree with this as well. I don't think Linux will ever be easy enough that I could recommend it to an elderly neighbor. I hope to be proven wrong, though.
What frustrates me about this particular moment is that at the same time Windows is getting worse, I feel that OS X is _also_ getting worse. This _is_ an opportunity for Apple to put a big dent in Windows market share.
> I don't think Linux will ever be easy enough that I could recommend it to an elderly neighbor.
The only reason I wouldn't do this is because that elderly neighbour wouldn't be able to install Linux and might not have any obvious place to get support from. Where can Grandma go to get support for her Linux laptop, even if she's willing to pay?
However, in a world where they can buy a laptop with Linux preinstalled and receive support from the same shop they bought it from if they do run into problems, then absolutely I would (not that that support is going to be great, but then they're at least no worse off than they were when they need support with Windows or a Mac, and I imagine they'll run into less problems on Linux than on Windows, given their use cases are likely to be very narrow and simple, i.e. web browser, e-mail, maybe simple office stuff).
> What frustrates me about this particular moment is that at the same time Windows is getting worse, I feel that OS X is _also_ getting worse. This _is_ an opportunity for Apple to put a big dent in Windows market share.
Aye, I agree. MacOS has been getting a bigger slice of the pie, but it's hard to ascribe what's the main cause, and to what extent each cause is contributing. We got the M chips being ungodly good (even the M1 is still serviceable, and damn right affordable even at this point), Windows growing worse, but the laptop market is also contracting, with a steady stream of people leaking out, saying 'screw this, I'll just use my phone or tablet. I don't need a PC for anything anymore.'.
All the casuals I know use a Mac for a laptop because they want something simple and functional, and Macs do that job, but they keep doing that job worse and worse. Everybody else casual might have a Windows laptop, but barely ever use it. The rest are gamers and power users, and thus need a proper machine and can't stick to a phone and tablet.
Apple could attract from the groups who would otherwise be done with non-phone/tablet computing, but their offering is growing weaker and weaker.
FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". I like that characterization b/c it seems more precise than "almost everywhere", but I've heard both.
I mention that because when I read the first footnote, I thought this was a mistake:
> boundedness alone ensures the subinterval infima and suprema are finite.
But it wasn't. It does, in fact, insure that infima and suprema are finite. It just does NOT ensure that it is Riemann integrable (which, of course the last paragraph in the first section mentions).
Thanks for posting. This was a fun diversion down memory lane whilst having my morning coffee.
If anyone wants a rabbit hole to go down:
Think about why the Dirichlet function [1], which is bounded -- and therefore has upper and lower sums -- is not Riemann integrable (hint: its upper and lower sums don't converge. why?)
Then, if you want to keep going down the rabbit hole, learn how you _can_ integrate it (ie: how you _can_ assign a number to the area it bounds) [2]
[1] One of my favorite functions. It seems its purpose in life is to serve as a counter example. https://en.wikipedia.org/wiki/Dirichlet_function
[2] https://en.wikipedia.org/wiki/Lebesgue_integral