RC high pass Circuit Step Input why C Initially acts as SS
From the professor's explanation you know that the voltage across the capacitor is proportional to the integral of current wrt time, so for the voltage to change  instantly, the current would have to be infinite.

Unless you have one if those imaginary circuits with no resistance or inductance in a loop, that can't happen. In your case you have the R of the RC, so the current is limited to input voltage divided by R.