IPA does not have explicit different written display between two monophthongs and one diphthong — a bad symbol choice if any languages have a double vowel sound.

This statement is untrue, and therefore the conclusion following the dash is unjustified.

I’m going to front-load the solution here before explaining its parts. The bottom line is that you have at least six possible ways in IPA of distinguishing adjacent monophthongs that therefore fall into distinct syllables from a diphthong of the same vowels:

1. Use an inverted breve diacritic (meaning “non-syllabic”) beneath the IPA symbol for the vowel that is acting as a semi-vowel instead of a full one for the three phonemic diphthongs in English: ⟨aʊ̯⟩ from mouth, ⟨aɪ̯⟩ from price, and ⟨oɪ̯⟩ from choice.
   
   You can also use those for non-phonemic diphthongs like ⟨eɪ̯⟩ from face, ⟨ou̯⟩ from goat, ⟨ɪi̯⟩ from see, ⟨ʉu̯⟩ from goose, ⟨ʌi̯⟩ from tight, and ⟨eə̯⟩ from tan. However, these are not phonemic, and in many dialects they become monophthongs or different diphthongs than the ones I’ve shown here. For example: “In Scottish, Upper Midwestern, and California English, /eɪ̯/ and /oʊ̯/ are monophthongal [eː] and [oː].” For that matter, even the phonemic diphthongs can be realized as phonetic monophthongs in some dialects: “In several American dialects such as Southern American English, /aɪ̯/ becomes monophthongal [aː] except before voiceless consonants.”
   
   You should probably avoid confusing learners with highly variable and fairly complicated phonetics, concentrating on phonemics alone, the way Kenyon and Knott do further below.

2. Use an inverted breve over both vowels to show that they make up just one syllable, so ⟨a͡ʊ⟩ for mouth, ⟨a͡ɪ⟩ for price, and ⟨o͡ɪ⟩ from choice.

3. Use a superscript for the semi-vowel, so ⟨aᶷ⟩ for mouth, ⟨aᶦ⟩ for price, and ⟨oᶦ⟩ from choice. You could also write those as semi-consonant superscripts, so ⟨aʷ⟩ for mouth, ⟨aʲ⟩ for price, and ⟨oʲ⟩ from choice.

4. Write the semi-vowel component as the corresponding semi-consonant, as in so ⟨aw⟩ for mouth, ⟨aj⟩ for price, and ⟨oj⟩ from choice for the three phonemic diphthongs. Kenyon and Knott replace ⟨j⟩ with ⟨y⟩ for those last three because that letter is more apt to make native speakers think of the right sound than the [j], which is never spelled that way in native words.
   
   In Spanish phonology, this strategy of rewriting semi-vowels as semi-consonants is observed only for rising diphthongs, not for falling diphthongs, because in Spanish any adjacent monophthongs that could theoretically fuse to become diphthongs mandatorily do so in all cases, unlike in English.
   
   As you see they just use the semi-consonants as needed for the rising diphthongs, much like Kenyon and Knott do for the falling ones as well.

5. If you won’t do any of those, then you can also separate adjacent monophthongs with a centered or baseline dot like /ko·ɪnˈsaɪd/ if there is no stress marker between them the way there is in /koˈɪnsɪdəns/.

6. At last recourse, you can even put a space between adjacent monophthongs, so /ko ɪnˈsaɪd/.

So it is simply not true that you cannot use IPA for languages with consecutive monophthongs. You certainly can: you just need to distinguish that case from the diphthong case, and IPA provides you with many ways of doing that.

IPA even lets you represent the hiatus that occurs in the eel using a glottal stop, /ðiʔ ˈil/. Now even if you do not notice the space, you’ll realize that you stop the glottis there.

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References

IPA does indeed explicitly differentiate the written display of diphthongs. From Wikipedia’s IPA article:

Diphthongs

Diphthongs are typically specified with a non-syllabic diacritic, as in ⟨uɪ̯⟩ or ⟨u̯ɪ⟩, or with a superscript for the on- or off-glide, as in ⟨uᶦ⟩ or ⟨ᵘɪ⟩. Sometimes a tie bar is used, especially if it is difficult to tell if the diphthong is characterized by an on-glide, an off-glide, or is variable: ⟨u͡ɪ⟩.

Notes

⟨a⟩ officially represents a front vowel, but there is little distinction between front and central open vowels, and ⟨a⟩ is frequently used for an open central vowel. However, if disambiguation is required, the retraction diacritic or the centralized diacritic may be added to indicate an open central vowel, as in ⟨a̠⟩ or ⟨ä⟩.

In the English section of the Wikipedia article on diphthongs, we see more examples of using the special “non-syllabic” diacritic,⟨ ̯ ⟩ (whose code point is U+032F COMBINING INVERTED BREVE BELOW) to show diphthongs in English:

English

In words coming from Middle English, most cases of the Modern English diphthongs [aɪ̯, oʊ̯, eɪ̯, aʊ̯] originate from the Middle English long monophthongs [iː, ɔː, aː, uː] through the Great Vowel Shift, although some cases of [oʊ̯, eɪ̯] originate from the Middle English diphthongs [ɔu̯, aɪ̯].

Notice how all those use the inverted breve to show a non-syllabic semi-vowel, the off-glide at the end of a falling diphthong.

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_{I am leaving the rest of this here because I believe it will be useful to future visitors to our site.}

This depends in part on how broad or loose the transcription used is, and whether the dot is ever used to represent a syllabic boundary as in [ɔ.ɪ] or with a special diacritic for a non-syllabic vowel like in [oɪ̯]. Here I am using a careful, narrow phonetic representation of actual sounds using square brackets.

These are not phonemic transcriptions, however, and therefore would probably just confuse learners: learners need to train their brains to think about English phonemics the way native speakers do.

Phonemic Systems

There are certain broad phonemic systems where something like coincide might be represented as somewhat close to how coin-side might be represented. It all depends how careful the system’s designer was.

However, systems like that tend to represent the phonemic diphthong in coin using symbols that are different from those used for those two phonemes when they are separate monophthongs. Here’s an example of such a system courtesy of Professor Lawler:

In the referenced phonemic system shown in that document originally from Kenyon and Knott as transmitted by Professor Lawler, they use /ay/, /aw/, and /oy/ for English’s three phonemic diphthongs. That uses the /y/ for IPA [j] because there is no need for IPA [y] when representing English phonemes. It also retains the look of the regular, more familiar alphabet, which is of some help here.

Comparing coincide with coin-side

Let’s think about the pair of somewhat similar sounding words coincide and coin-side. Would a learner confuse the adjacent monophthongs of coincide with the phonemic diphthong of coin-side? They shouldn’t, as under the phonemic system just presented, those two would work out to this:

• coincide /koɪnˈsayd/
• coin-side /ˈkoynsayd/

Those are pretty obviously not the same thing, so these are not at risk for confusion using phonemics.

But about if we were using phonetics, though, instead of phonemics? Could there now be a problem?

The answer to that here as before again depends on how careful your phonetics are, but if you do it right, there should still be no risk of confusion. A narrow phonetic transcription might run more like:

• coincide [ˌkʰoʷᵻnˈsäɪ̯d]
• coin-side [ˈkʰoɪ̯̃ŋsäɪ̯d]

which again are reasonably distinct. They’re also a bear to key in, which is yet another reason why we don’t use narrow phonetics that often.

I recommend use a broad phonemic transcription such as this one when teaching English pronunciation. It does a good job of focusing on what the English-speaking brain focuses on, and of disregarding the narrow phonetic variations seen in the many dozens of allophones.

Phonetic Systems

For the remainder of this post, I will use narrow phonetic transcriptions between square brackets instead of broad phonemic transcriptions between slashes.

I don’t believe the sort of thing you’re talking about ever happens across word boundaries in English. The first of these will never become the second:

• The saw is all right.  [ðəˈsɔ.ᵻz.ɔɫˈɻäɪ̯ʔ]
• The soy’s all right.    [ðə ˈsoɪ̯z.ɔɫˈɻäɪ̯ʔ]

Connected speech can be hard to represent for a variety of reasons, but I don’t believe that bisyllabic [ɔ.ɪ] with two monophthongs in hiatus across two syllables can ever fuse into the tautosyllabic diphthong [oɪ̯] in just one. I suppose it may happen in dialects I’m not familiar with.

“Naïve koalas never coöperate with extraördinary knives”

I’m assuming you only care about the three phonemic diphthongs, which are all falling diphthongs. If not, then sometimes you can get one of the two vowels reducing to a glide and producing a rising diphthong, at the cost of that hiatus and therefore of a syllable.

• koala [kʰoˈɑlə] > [kʰʉˈwɑlə] > [ˈkʰwɑlə] > [ˈkʰʷɑlə]
• cooperate [kʰoʷˈɑpəˌɻeʲt] > [kʰʉˈwɑpəˌɻeʲt] > [ˈkʰʷɑpɻʷëʔ]

Continuity Hypothesis

In the case of your word continuum, there is a [w] glide there blocking fusion but it may not be written in all transcriptions.

• Phonemic: /kənˈtɪnjuəm/ or more explicitly /kənˈtɪn·ju·əm/
• Phonetic:    [kʰə̃nˈtʰɪ̃n·juu̯·ə̃m] or [kʰn̩ˈtʰɪ̃·ɲuw·m̩]

There’s already a phonetic triphthong in the ⟨njuu̯⟩ syllable to the following ⟨ə̃m⟩. I don’t know any argument that could be made in which fusing those would ever be possible, let alone that doing so could produce a tautosyllabic phonetic tetraphthong or pentaphthong or whatever the heck that would make it. I don’t know any way to merge that into the syllabic consonant at the end.

That’s the same reason why naïves [näɪ̯ˈjivz] can never become knives [näɪ̯vz], because here a [j] glide gets in the way.

So often either a [j] or a [w] glide blocks any possibility of fusion, providing a semi-consonantal pivot-point between two syllables. Unlike in the narrow phonetic transcriptions I’m using here, these little glides are seldom written in broad phonemic transcriptions because native speakers always apply them anyway.

In a few other cases, one of the adjacent monophthongs will simply be dropped, the way we also hear in extraordinary [ɛkˈst͡ʂɻʷo˞dəˌneɻi] , where there is no longer an [ə] syllable at the end of extra before starting on the [o] of ordinary the way you would expect if it had been written extra-orginary or extraördinary.