Is that a shocker to you?
First, let me explain an infinitive. It’s a verb form in English that uses the word “to” with the root form of the verb. The result, a kind of verbal, is called an infinitive. It can function as a noun, as in “To err is human; to forgive, divine.” There, it’s the subject of each clause. I wrote about two kinds of verbals here. This one, the infinitive, is the one I didn’t cover there. In this post I’m focusing on the false belief that there is such a thing as a “split infinitive.” I’m not explaining the verbal form.
So, all right. We have a two-word formation, like “to lose.” That “to” is not grammatically part of the infinitive. Rather, in Oxford English Grammar Greenbaum calls it a subordinator, while Huddleston/Pullum in their Cambridge Grammar of the English Language prefer “the infinitival to.” The grammatical part is that root form, “lose.” No grammatical rule says that other words can’t come between that “to” and the root form. Some folks in the 19th century got it into their heads, though, to force Latin grammar rules onto English, so we ended up with the specious nonrule against “split infinitives.” But here’s the thing: In Latin (and other languages as well, like French), it’s impossible to “split” an infinitive because that form is a single word. There’s nothing to separate, nowhere to intervene with another word.
Don’t tie yourself into knots trying to keep that “to” with that root verb. It’s not wrong. There’s no rule saying you can’t write “He wanted to just be left alone.” (The question of whether that “just” is worth keeping is a separate one, and I’m not touching it here. One could as easily replace it with “simply,” if that makes it more palatable to the reader.) And if you see it written that way, you’re under no obligation to “correct” it. There’s nothing to correct, from a grammatical standpoint.
If you’re up for a bit of research, I’ll recommend getting your eyes/hands on the aforementioned grammar texts. Personally, I have the student guide version of Cambridge because the full edition was out of my price range. In Greenbaum, the “split infinitive” is relegated to a chapter note in the section at the end of the text. (It didn’t even earn space in the main text. Only a note.)
Grammargeddon! 
English is a language of special cases and exceptions, so there are some verbs for which the infinitive (without the “to”) consists of more than one word.
An example might be “to point out”. Splitting the two parts of the infinitive in this case would be very strange and likely incorrect. Consider “to point appropriately out” vs “to point out appropriately” or even “to appropriately point out.”
Plenty more examples at https://learnenglish.britishcouncil.org/grammar/english-grammar-reference/multi-word-verbs
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What you’re talking about there, Frank, is also called a “phrasal verb.” The rules are the same regarding a word coming between the “tp” and the rest of the verb, in your example “point out.” We can say “to loudly point out” or “to point out loudly,” but because phrasal verbs are very seldom split (I can’t think of an example where one is that doesn’t sound bizarre to the AmE speaker’s/writer’s ear), we cannot say “to point loudly out.” Let’s be cautious about muddling what’s meant by a “split infinitive” (which, grammatically, does not exist in English because that “to” is not part of the verb) and a poorly used phrasal verb.
The fact remains that there is no grammatical rule about a word intervening between the “to” and the infinitive form. What I explained just now works with any phrasal verb and an appropriate modifier: sit down slowly/slowly sit down, stand up quickly/quickly stand up, pick up roughly/roughly pick up, set down gently/gently set down, etc. We wouldn’t say “he sat slowly down” under normal circumstances, but there’s a certain poetic sound to that phrasing and it could be appropriate in some registers. The same isn’t true, at least to my ear, with “pick roughly up.” That’s just strange. Many “rules” like this one are more accurately termed “conventions,” arrived at over time by many users. If “point out appropriately”” sounds bad to you, by all means don’t use it. To me, it sounds fine. (Here I’ll suggest Longman’s Dictionary of Phrasal Verbs, for those who want to learn more. I also wrote about them here at the blog. Just search for “phrasal verbs.”)
However, there is no such animal as a split infinitive in English grammar, because that “to” is not grammatically part of the verb. Don’t take my word for it; go to Huddleston and Pullum, or to Greenbaum (Oxford and Cambridge, respectively).
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