Is there an app or any way at all to create a screencast on an iPhone?

There was an app that did this in the app store, but it's no longer available. It's called Display Recorder. Later it changed its name to DispRecorder. I have it and it works for recording most things on your iPhone screen. Some apps don't show up in the recording, though. It was available briefly last year, here's an article about it: http://ipadinsight.com/ipad-apps/display-recorder-ipad-app-is-back-with-an-odd-new-name-screen-recording-on-the-ipad-demo/ As others have mentioned, my preferred method is to mirror my iPhone or iPad screen on my Mac and use Screenflow to capture from there. I use a Mac app called Reflector: http://www.reflectorapp.com/ (also works on Windows) Your iOS device and your computer need to be on the same wifi network, so this works great for me at home. But when I was at MIT (where I used to work), I could not get it to work because their network is divided into subnets and you can't force your devices to be on the same one. I found that to be true at other institutions I've visited, too. The workaround is to create an ad-hoc network for your computer and iOS device, but then you don't have external internet access, which may be part of what you want to show in the app! So I do all my video creation at home with Reflector and Screenflow on my Mac desktop. Good luck!

Do you have access to Xcode? If so there is a way to inject a code inside the simulator then after running "your" app, ie you must have the source code for it,  you can record with high fidelity whatever you do. If you answer that you have Xcode and you have an app that can run inside the simulator then I can pass you more instructions.

I wrote a blog post on how to do this - here you go! http://www.edsocialmedia.com/2014/01/make-video-screen-capture-ipad-iphone-without-jailbreaking/