I am having difficulties with this case. I know that υπήρχα is past continuous. I say υπήρξα frequently and get corrected to υπήρχα. It seems I existed would always be past continuous but obviously it isn't. Can someone give me some examples for both?
Υπήρξα is mostly used with the sense "I was". Obviously, it can't have literal sense for as long as you are alive. Example: Υπήρξα πρόεδρος του ορειβατικού συλλόγου. (I was ... (but I am not any more)). Υπήρχα is less common (in 1st person), unless in a negative scheme: Δεν θα υπήρχα αν ...
Υπήρχαν παιχνίδια στο πάτωμα. (There were toys on the floor. => the toys were lying on the floor)
Υπήρχαν περίοδοι/στιγμές που ένιωθα θλίψη. (There were times/moments I wasfeeling blue. => the emphasis on the continuity of the feeling in time)
Υπήρξαν περίοδοι/στιγμές που ένιωσα άσχημα. (There were times/moments I felt bad. => the emphasis on the recurrence of a feeling that was not permanent)
Υπήρξα πρόεδρος του... (I served as president of...)
As sotos wrote, υπήρχα (in the 1st person) is less common:
Ένιωθα πως απλώς υπήρχα, δεν ζούσα πραγματικά. (I felt I merely existed [I was just being], I was not literally living.)
Neither of these two sentences is really idiomatic.
The first one is possible, but one would be much more likely to say Υπήρχαν πέντε άτομα στο πάρτι, and even that would sound somewhat strange to me. Υπήρχαν πέντε πιάτα στο τραπέζι would be OK, though.
The second one is completely unidiomatic. The aorist υπήρξα is only used to mean "existed", or, followed by a predicate noun or adjective, "was" -- with a strong implication of "no longer is".
Υπήρξε ο Χριστός; = Did Jesus Christ exist?
O Κανάρης υπήρξε και πρωθυπουργός = Kanaris was also prime minister for a time.