"päivänjatko" - continuation of the day
You wouldn't really say this to wish someone a good day as they walk out the door in the morning. You could use "Hauskaa päivää!" or "Mukavaa päivää!" for that.
"Hyvää päivänjatkoa!" is something you might hear cashiers say, for instance, to which you could reply "Kiitos samoin!" ("sama" - the same)
I was curious to know how English influenced Finnish, since you said: ¨All in all wishing hyvää päivänjatkoa is quite a new expression, from this century, likely because of influence of English¨.
I would add that, once the day is started, the remaining is always ¨the rest of the day¨. Even if you meet someone in the morning, then, he has ´the rest of the day´ in front of him.
Indeed it´s just an expression, so the ´jatkoa´ does not really matter much in my opinion.
I ask you to clarify me KIITOS from what I read, there is Vowel Harmony in the Finnish language and from what I understand, "ä" and "ö" are front vowels and "a" and "o" are back vocals and they almost never occur in the same word, except when they are compound words, so is "päivänjatkoa"a compound word?
Päivänjatkoa is indeed a compound word. The parts are:
- päivä : a day
- n : the genitive marker, the genitive is a common "glue" case with which words are put together(*)
- jatko : a continuation
- a : the partitive marker, since the phrase can be considered to be shortened from (Toivotan) hyvää päivänjatkoa! : (I wish you) a good continuation of the day!
*: The genitive case in English (my, your…) is strictly of possession. Because of the Romance, mostly French, language influence English uses in many cases the of-construction (the colour of the house vs. the house's colour). This of-construction can denote other things, for instance location, as the Finnish genitive case does, kaupungin keskellä : in the middle of the town.
It took me some time to get the joke since you wrote with a's instead of ä's. It would be a quite clever joke except the ending -ja/jä denoting the person performing the act needs to be attached to a verb stem ("the act") to be meaningfull. Anyway +1 for the effort!