Is it possible to merge two or more videos in real-time like this? Is it possible to play video online that's made of two or more video files?

Here's what I mean.

My site is hosted on Linux/Apache/PHP server. I have video files in FLV/F4V format. I can also convert them to other available formats if necessary. All videos have same aspect ratio and other video parameters.

What I want is to have online video player that plays video composed of multiple video files concatenated together in real-time, i.e. when user clicks to see a video.

For example, user sees video titled "Welcome" available to play. When he/she clicks to play that video, I take video files "Opening.f4v", "Welcome.f4v" and "Ending.f4v" and join/merge/concatenate them one after another to create one continuous video on the fly.

Resulting video looks like one video, with no visual clues, lags or even smallest observable delay between video parts. Basically what is done is some form of on-the-fly editing or pre-editing, and user sees the result. This resulting video is not saved on the server, it's just composed and played real-time.

Also, if possible, user shouldn't be made to wait for this merging to be over before he/she sees resulting video, but to be able to get first part of the video playing immediately, while merging is done simultaneously.

Is this possible with flash/actionscript, ffmpeg, html5 or some other online technology? I don't need explanation how it's possible, just a nod that it's possible and some links to further investigate.

Also, if one option is to use flash, what are alternatives for making this work when site is visited from iphone/ipad?
DASH HTML5 video streaming already works by getting the player to load video in chunks that are usually about 10 seconds long.  You should be able to cook something up that gets the player to display your segment seamlessly, by giving it a DASH-like list of videos.

You might need to remux your video segments into .mp4, if some web browsers choke on the flv container format.  (assuming that's the container for f4v, which I've never seen before.)