I broke my iphone 5 glass, can I just buy new glass cover for it?

My girl recently dropped her iPhone 5 and the glass is cracked on the front. The touchscreen still works great, but the cracked glass is obviously very annoying and will undoubtedly get worse with time. Can I just replace it with a generic outer glass or do I also have to replace the digitizer touch function when I replace the glass on it?

Yes you can.