Why can't a causal digital filter have an infinitely sharp transition between the passband and the stopband?
I don't have a concrete proof for this one. However, I can tell you this... 
Consider a perfect low pass filter. The time domain representation is a sinc. And for any system to have a sharp transition band, a base signal has to be multiplied with a rectangular waveform in the frequency domain. Which implies that, the time domain signal of the same has to be convoluted with a sinc in time domain. We know that sinc function is a non causal signal. 
Hence proved.