Learn Hardcore Danish: Det er læreren, der fremhæver, hvor vigtigt det er at have både tid til job og til fritid. - It is the teacher who emphasizes how important it is to have time for both work and free time.

Læreren fremhæver, hvor vigtigt det er …
= The teacher emphasizes how important it is … (neutral statement)
Det er læreren, der fremhæver, hvor vigtigt det er …
= It is the teacher who emphasizes how important it is … (focus on the teacher, maybe in contrast to others)

So the cleft form answers an implicit question like:
Who is it that emphasizes this? → Det er læreren, der …

What is the function of der in Det er læreren, der fremhæver …? Is it the same der as in der er (“there is”)?

No, it is not the same function.

In Det er læreren, der fremhæver …, der is a relative pronoun, similar to English who / that:

Det er læreren, der fremhæver …
≈ It is the teacher *who emphasizes …*

In Der er en lærer i lokalet, der is an expletive / dummy subject, like English there in There is a teacher in the room.

So:

relative der = who/that (introduces a relative clause about læreren)
dummy der = there (in der er / der var / der bliver etc.)

Why is there a comma before der in Det er læreren, der fremhæver …?

The comma marks the beginning of a relative clause:

Main clause: Det er læreren
Relative clause: der fremhæver, hvor vigtigt det er …

In standard written Danish, you normally put a comma before a relative clause introduced by som or der:

Det er manden, som bor derovre.
Det er læreren, der fremhæver …

Some people (following the so‑called “new comma”) may omit some commas in more informal writing, but the comma before a relative clause is still very common and considered correct and clear.

Could you also say Det er læreren, som fremhæver … instead of der fremhæver? Is there a difference?

Yes, you can say:

Det er læreren, som fremhæver, hvor vigtigt det er …

In modern Danish, som and der are often interchangeable as relative pronouns referring to the subject:

Det er læreren, der fremhæver …
Det er læreren, som fremhæver …

Both are correct. Some nuances:

der is slightly more common and neutral in spoken Danish in this position.
som can sound a bit more careful or formal in some contexts, but the difference here is small.

In many cases, you can freely choose between them when they refer to the subject (the one who does the action).

Why is it hvor vigtigt det er and not hvor vigtig det er? What does vigtigt agree with?

Vigtigt is the neuter form of the adjective, and it agrees with det.

In this construction, det is a dummy subject, and the adjective has to match it in gender and number:

Neuter singular: vigtigt
- Det er vigtigt. – It is important.
Common gender singular: vigtig
- Sagen er vigtig. – The matter is important.

In your sentence, the inner clause is:

hvor vigtigt det er at have både tid til job og til fritid
= how important it is to have both time for work and for free time

Because the subject is det, you must use the neuter form vigtigt.

Why is the word order hvor vigtigt det er and not hvor vigtigt er det?

This is an embedded (subordinate) clause, not a direct question.

Direct question: Hvor vigtigt er det?
– Verb comes right after the question word (V2 word order).
Embedded clause: Jeg ved, hvor vigtigt det er.
– Verb comes later (subject before the verb): … det er.

In your sentence, hvor vigtigt det er at have … is part of what the teacher emphasizes. It’s like an indirect question or a content clause:

læreren fremhæver, hvor vigtigt det er …
= the teacher emphasizes how important it is …

So it needs subordinate clause word order: [hvor vigtigt] [det] [er] …, not [hvor vigtigt] [er] [det] ….

What exactly does fremhæver mean here? Is it like “highlight”, “emphasize”, or something else?

Fremhæve means roughly to highlight / to emphasize / to point out.

læreren fremhæver, hvor vigtigt det er …
= the teacher emphasizes / points out how important it is …

It is a regular verb:

infinitive: at fremhæve
present: fremhæver
past: fremhævede
past participle: fremhævet

It usually takes an object (explicit or implicit):

Hun fremhæver fordelene. – She emphasizes the advantages.
Han fremhæver, hvor vigtigt det er … – He emphasizes how important it is …

Why do we use både … og … (både tid til job og til fritid)? Is it necessary to have both words?

Yes, in Danish the pattern både … og … works as a pair and corresponds to English both … and ….

både X og Y = both X and Y

Here:

både tid til job og til fritid
= both time for work and (time) for free time

If you leave out både, you just get a simple og:

tid til job og til fritid
= time for work and for free time (no special emphasis on “both”)

With både, the sentence underlines the idea that both aspects are important, not just one of them.

Why is it tid til job og til fritid and not tid til job og fritid? Do we have to repeat til?

You do not have to repeat til, but it is quite common and fully natural.

Both are acceptable:

tid til job og fritid
tid til job og til fritid

Repeating til:

Makes the parallel structure clearer:
- til job / til fritid
Can give a slightly more balanced, rhythmical feel.

Leaving it out (til job og fritid) is also correct and probably a bit more compact. In this particular sentence, repeating til sounds very natural, but it’s mostly a stylistic choice, not a grammar rule.

Why is job used here and not arbejde? Could you say tid til arbejde instead of tid til job?

You could say tid til arbejde, but there is a nuance:

job often refers to your occupation / position (your job as a concrete thing in life).
arbejde is more the work itself, the activity or tasks.

In this kind of everyday phrase:

tid til job suggests “time for your job / your employment”.
tid til arbejde might feel a bit more like “time for doing work” in a general sense.

Both are understandable, but tid til job fits the contrast with fritid (free time) very naturally: time for your job and time for your free time.

Why is it til fritid and not til fritiden? What is the difference?

The difference is between indefinite and definite:

fritid = free time (in general, as a concept)
fritiden = the (specific) free time

In your sentence, we are talking about free time in general as a life balance:

… at have både tid til job og til fritid.
= to have both time for work and for free time (in general).

If you said til fritiden, it would sound more like a specific block of free time that has already been defined in the context, which doesn’t fit as well in this general statement about life balance.

Could you leave out det er and just say Læreren fremhæver, hvor vigtigt det er at have …? How would the meaning change?

Yes, that is perfectly grammatical:

Læreren fremhæver, hvor vigtigt det er at have både tid til job og til fritid.

The difference is mainly in focus:

Læreren fremhæver …
– Neutral: just stating what the teacher does.
Det er læreren, der fremhæver …
– Emphasizes that it is the teacher (and maybe not someone else) who does this.

So the cleft form with Det er … der … adds focus on læreren as the key person, while the simple form is more neutral.