Oh! My bad! I completely missed that the functions were continuous (it isn’t required for 🍊 to be a metric)

Careful ⚠️ there is not guaranteed to be an element such that |🍎(x) - 🍇(x)| is maximized. Consider 🍎 (x) = x if x < 3, 0 otherwise. Let 🍇 (x) = 0, and let the domain be [0, 4]. Clearly, the sup(|🍎 (x) - 🍇 (x)| : x ∈ [0, 4]) = 3, but there is no concrete value of x that will return this result. If you wish to demonstrate this in this manner, you will need to introduce an 🐘 > 0 and do some pedantic limit work.

Anyway, to prove this is a metric we must prove that it satisfies the 4 laws of metrics.

1. The distance from a point to itself is zero. 🍊 (🍎, 🍎) = 0

This can be accomplished by simply observing that |🍎 (x) - 🍎 (x)| = 0 ∀x ∈ [a,b], so its sup = 0.

2. The distance between any two distinct points is non-negative.

If 🍎 ≠ 🍌, then ∃x ∈ [a,b] such that 🍎 (x) ≠ 🍌 (x). Thus for this point |🍎 (x) - 🍌 (x)| > 0 and the sup > 0.

3. 🍊 (🍎, 🍌) = 🍊 (🍌, 🍎) ∀(🍎, 🍌) in our space of functions.

Again, we must simply apply the definition of 🍊 observing that ∀x ∈ [a,b] |🍎 (x) - 🍌 (x)| = |🍌 (x) - 🍎 (x)|, and the sup of two equal sets is equal.

4. Triangle inequality, for any triple of functions (🍎, 🍌, 🍇), 🍊 (🍎, 🍌) + 🍊 (🍌, 🍇) ≥ 🍊 (🍎, 🍇)

For any (🐁, 🐈, 🐕) ∈ ℝ³ it is well known that |🐁 - 🐕| ≤ |🐁 - 🐈| + |🐈 - 🐕|, (triangle inequality of absolute values).

Further, for any two functions 🍍, 🍑 we have sup({🍍 (x) : x ∈ [a, b]}) + sup({🍑 (x) : x ∈ [a, b]}) ≥ sup({🍍 (x) + 🍑 (x) : x ∈ [a, b]})

Letting 🍍 (x) = |🍎 (x) - 🍌 (x)|, and 🍑 (x) = |🍌 (x) - 🍇 (x)|, we have the following chain of implications:

🍊 (🍎, 🍌) + 🍊 (🍌, 🍇) = sup(🍍 (x) : x ∈ [a, b]}) + sup({🍑 (x) : x ∈ [a, b]}) ≥ sup({🍍 (x) + 🍑 (x) : x ∈ [a, b]}) ≥ sup({🍎 (x) - 🍇 (x)| : x ∈ [a, b]) = 🍊 (🍎, 🍇)

Taking the far left and far right side of this chain we have our triangles inequality that we seek.

Because 🍊 satisfies all four requirements it is a metric. QED.

QED stands for 👸⚡💎, naturally

What happened to 🍇 and 🍍?

Johnny Mnemonic

Why is “gives free tech support,” an expectation of being a good person? I spent 7 years of my life doing tech support professionally.

a) I never want to do it again, because honestly fuck computers, but b) If I have to, I’m certainly not going to do it unless I already like you or you are paying me.

Figure your own shit out. I have a life to live and I don’t have infinite free time and energy to help you unfuck whatever stupid bullshit happened to your computer. Especially if you’re being entitled about it.

You might find this interesting: https://citationsneeded.libsyn.com/ep-152-hallmark-christmas-movies-and-the-cozy-conservative-nostalgia-machine

Transcription: https://citationsneeded.medium.com/episode-152-hallmark-christmas-movies-and-the-cozy-conservative-nostalgia-machine-10b6b848ab2a

I mean, by pushing conservative media narratives and moving the Overton window to the right, you could argue that Hallmark is in part responsible for our illegal ICE deportations to El Salvadorian torture prisons, the rise in hate crimes against transpeople, and our future war with Iran.

But maybe an extra white lady got killed too.

Are you kidding? Do you not remember? That is exactly what a pokemon journey is for! Now put on a baseball cap, grab your Bulbasaur, and get out there and be the very best like no one ever was.

To be fair, the last time I saw this image was probably either in the 90s or early 2000s

This is the highest resolution of this picture I’ve ever seen.

Bloom County was so great.