Yes, it is a begonia. It's looks similair to mine which is, B. Persian Brocade (eyelash begonia) Only the leaves on yours look a little pale and smaller. Did it ever look like this?
lol, there are SO many!! i didn't hazard a guess as to which particular one...it's definitely not rex or a strawberry (the two that i have)... the leaves are very close to the strawberry, though.
It is very similar to Begonia hernandioides. The problem is these plants have been hybridized so many times they often appear similar to the species but may not truly be what they appear.
I grow this Begonia and a fair number of others. We used to have tons of Begonia species when we lived in Miami but not as many now. Begonia hernandioides has been used to create a fair number of hybrids so there is no guarantee the one in the photo is that species. But I'd say it is very close, if not the species. But I will says very quickly, my knowledge of Begonia species is limited. I have the email addresses of several officers of the Begonia Society and often ask one of them for an ID.
There are so many different begonias it can make your head spin if your not a collector, and so my knowledge of begonia's is limited as well. I own four, and maybe know of 6!
There is a grower in Miami we used to visit quite often. The place is called Palm Hammocks Orchid Estate. I'm not sure if they still do, but at one time they had over 500 species and hybrids of Begonia on display and for sale. Janice used to love the things and would often bring home another one. And that is where I bought the original Begonia hernandioides we grew. If my memory is correct, that one didn't make the move from Florida but someone here gave me a replacement. The owner told us one time he had personally grown over 800 different types. He was also an officer of the local Begonia society and I believe an officer of the national group as well. I do know they often had big meetings of Begonia growers on the property.