Inspired by this question: Multi purpose Arcane Focus?

The premise of the other question is about using the Rod of the Pact Keeper as a focus, and although it implies it can be used as a focus indirectly, I cannot see in the description of the magic item (DMG, pg. 197) anywhere that says it can be used as a focus, nor in any generic text about rods (DMG, pg. 139).

Rod of the Pact Keeper

Rod, uncommon (+1), rare (+2), or very rare (+3) (requires attunement by a warlock)

While holding this rod, you gain a bonus to spell attack rolls and to the saving throw DC of your warlock spells. The bonus is determined by the rod's rarity.

In addition, you can regain one warlock spell slot as an action while holding the rod. You can't use this property again until you finish a long rest.

Given that it isn't explicitly said in the description, I assume it must be possible for all rods to be used as foci. RAW, can any rod be used as a focus?

No.

Magic items that can be used as a spellcasting focus, such as a hat of wizardry, say so explicitly:

This antiquated, cone-shaped hat is adorned with gold crescent moons and stars. While you are wearing it, you gain the following benefits:

• You can use the hat as a spellcasting focus for your wizard spells.

There's no general rule that says that any magic rod (or staff, or orb) has this property. The only general properties attributed to items of these types in the definitions of magic item categories in the DMG is:

Unless a staff’s description says otherwise, a staff can be used as a quarterstaff.

There's no mention of using any category of magic items as a spellcasting focus.

Jeremy Crawford has, however, stated that all wands do indeed have this property, even though it is not explicitly spelled out anywhere in the rules.

My interpretation is that a wand has no possible purpose other than as an aid to casting spells. Not all objects that can be constructed as spellcasting foci are so singular in their purpose; a wooden or metal rod can be a club, tool, or an axle; a staff can be a weapon or a walking-stick. That's why a quarterstaff costs two silver pieces, but a staff fashioned for use as an arcane focus costs twenty-five times as much.

It doesn't seem reasonable to assume that a utility magic item like an immovable rod, or an item clearly intended for martial purposes, like a rod of lordly might, is also constructed for use as a spellcasting focus.

Of course, a dungeon master might quite reasonably state that any given magic rod (or staff, or orb) is, in addition to its magical properties, also constructed to be used as a spellcasting focus, but that's not explicitly stated to be automatically true anywhere.

The Sage said wands can while the book does not mention it.

Since the only hint (apart from their names) that magical rods, wands and staves can be used as foci are their tags rod, wand and staff that classify them as such, then we look at their tags descriptions at pages 139-140 of DMG but as you previously said, there is no explicit information about their use as arcane foci.

However, according to Jeremy Crawford, any wand can be used as an arcane focus. I can't see a reason why it wouldn't also apply for rods and staves since they usually are also foci but none of them had it written on their descriptions.

@JeremyECrawford Can Wand of the War mage be an arcane focus? Doesn't specify, and while wands can be foci, it's not clear that ANY wand can

JC: Any wand can be used as an arcane focus.

The descriptions of DMG magic items are not exhaustive.

Marq commented:

An arcane focus is defined as "a special item ... designed to channel the power of arcane spells". Magic items of the "rod" category are "typically made of metal, wood, or bone ... about 2 or 3 feet long, 1 inch thick, and 2 to 5 pounds." Not all mundane rods of those dimensions are arcane foci (some may be, for example, cart axles, or crowbars); thus not all magic rods are. In contrast, all swords are weapons, by definition; it's not a special subcategory of swords that are made for attacking

First, I wouldn't call magic rods and staves something mundane; they are magical.

Second, the DMG magic items descriptions (p139-140) don't describe their every function, they do not specify their most basic functions since magic wands have nothing written there that they can be used as foci as weapons doesn't say that they can be used to attack. The same thing about rods and staves that doesn't explicitly say about their use as foci. RAW is the same to say that I can use a normal sword to cut someone but I can't use a Dancing Sword to do the same because isn't written that I can unless I make it fly while in my hand it doesn't have the capability to injure anyone. That would be unreasonable.

PHB also doesn't help.

There are no non-foci rods or wands described in the PHB. Would all rods and wands in existence be foci? That would also be unreasonable since it clearly says they are specially prepared to be used as such. If you imply that someone could make a non-foci rod with a log to support a broken table or a non-magical wand with silver and jade for a teacher to point at the blackboard because it's reasonable but isn't written in the book, then someone can make an ornamental sword that isn't a weapon and also isn't written there. Just because isn't written in the book doesn't mean it can't be done.

The hat of wizardry case.

Marq said in his answer:

Magic items that can be used as a spellcasting focus, such as a hat of wizardry, say so explicitly:

This antiquated, cone-shaped hat is adorned with gold crescent moons and stars. While you are wearing it, you gain the following benefits:

• You can use the hat as a spellcasting focus for your wizard spells.

Unlike wands, staves and rods, hats are not usually arcane foci. If a specific item that usually doesn't have such function now works as such, obviously it need to be stated in its description. That's the reason the hat of wizardry clearly have it noted there. Since wands, rods and staves are meant to be used as foci as much as weapons are meant to be used to attack, they didn't need to have it noted in their entries; it's their basic function, you just assume they do what they are supposed to do.