Το καθήκον μου είναι να φροντίζω την οικογένειά μου.

Because καθήκον is a neuter singular noun, and here it appears as a definite noun phrase: το καθήκον μου = my duty.

Greek uses the definite article much more often than English. So even though English says my duty without the, Greek naturally says το καθήκον μου.

Grammatically:

• το = neuter singular definite article
• καθήκον = neuter singular noun
• μου = my

So literally, it is something like the duty of mine, but the natural English meaning is simply my duty.

In Modern Greek, the usual possessive words μου, σου, του, της, μας, σας, τους normally come after the noun.

So Greek says:

• το καθήκον μου = my duty
• η οικογένειά μου = my family
• το σπίτι μου = my house

This is just the normal Greek pattern. A native English speaker often expects the possessive to come first, but Greek usually puts this short possessive form after the noun.

Because it belongs to two different nouns:

• το καθήκον μου = my duty
• την οικογένειά μου = my family

Each noun needs its own possessive marker. Greek does not normally let one μου cover both parts of the sentence from a distance.

So the repetition is completely normal.

Να introduces a verb clause. In many cases, it corresponds to English to before a verb:

• να φροντίζω ≈ to take care

But it is important not to think of να as exactly the same as English to. Modern Greek mostly does not use an infinitive the way English does. Instead, it uses να + verb form.

So:

• English: My duty is to take care...
• Greek: Το καθήκον μου είναι να φροντίζω...

This is one of the most basic patterns in Greek.

Because these two verbs belong to different clauses.

• είναι goes with το καθήκον μου
  → My duty is...
• φροντίζω refers to the speaker
  → ...that I take care of...

So the structure is roughly:

• Το καθήκον μου είναι = My duty is
• να φροντίζω την οικογένειά μου = to take care of my family

Greek shows the subject inside the να clause through the verb ending. The ending -ω in φροντίζω tells you the subject is I.

That is why the sentence has:

• είναι = is
• φροντίζω = I take care

This is an aspect question.

• να φροντίζω = ongoing, repeated, habitual, general action
• να φροντίσω = one whole action, or action viewed as a single event

In this sentence, the idea is a continuing responsibility: taking care of one’s family is not a one-time act. So να φροντίζω is the natural choice.

Compare:

• Το καθήκον μου είναι να φροντίζω την οικογένειά μου.
  My duty is to keep taking care of my family / to care for my family.

• Το καθήκον μου είναι να φροντίσω την οικογένειά μου.
  This could sound more like a specific task, mission, or one complete responsibility viewed as a whole.

Both are possible in some contexts, but the version with φροντίζω fits the general meaning best.

Because οικογένεια is the direct object of φροντίζω.

You take care of someone/something, so that noun goes in the accusative case in Greek.

Here:

• dictionary form: η οικογένεια = the family
• accusative singular: την οικογένεια = the family

So:

• φροντίζω την οικογένεια = I take care of the family

Then add the possessive:

• την οικογένειά μου = my family

So την is the feminine singular accusative article.

Because of the enclitic μου.

The noun οικογένεια normally has its main written accent on γέ. But when a word like this is followed by an enclitic such as μου, Greek often adds an extra written accent near the end:

• οικογένεια
• οικογένειά μου

This is a normal spelling rule in Greek. You will see the same thing in many similar phrases.

So the extra accent is not changing the meaning. It is just the correct written form when the enclitic μου follows.

Not completely. Greek word order is more flexible than English word order. The sentence you have is the neutral, natural order:

• Το καθήκον μου είναι να φροντίζω την οικογένειά μου.

But other orders are possible for style or emphasis. For example:

• Καθήκον μου είναι να φροντίζω την οικογένειά μου.
  More formal or literary.

• Να φροντίζω την οικογένειά μου είναι το καθήκον μου.
  More marked; it emphasizes the action.

So the original version is the best one to learn first, but Greek does allow some movement for emphasis.

Yes, it can, but the style changes.

• Το καθήκον μου είναι... = neutral, standard, everyday phrasing
• Καθήκον μου είναι... = more formal, elevated, or literary

A native speaker would very naturally say the full version with το. Omitting it is possible, but it sounds less neutral.

So if you are learning the standard pattern, keep:

• Το καθήκον μου είναι να φροντίζω την οικογένειά μου.