a) Discrete time :

             { By Commutative Property }

In order for a discrete  LTI  system to be causal, y[n] must not depend on x[k] for k > n. For this to be true h[n-k]'s corresponding to the x[k]'s for k > n must be zero. This then requires the impulse response of a causal discrete time LTI system satisfy the following conditions :

 

Essentially the system output depends only on the past and the present values of the input.

 Proof : ( By contradiction )

 Let in particular h[k] is not equal to 0, for some k<0

  {Refer the eqn. above}

So we need to prove that for all x[n] = 0,  n < 0, y[0] = 0

Now we take a signal defined as

 

 This signal is zero elsewhere. Therefore we get the following result :

 We have come to the result that y[0] 0, for the above assumption. our assumption stands void.  So we conclude that  y[n] cannot be independent of x[k] unless h[k] = 0  for  k < 0

 Note : Here we ensured a non-zero summation by choosing x[n-k]'s as conjugate of h[k]'s.