b) Continuous time :

 

 In order for a continuous  LTI  system to be causal, y(t) must not depend on x(v) for v > t . For this to be true h(t-v)’s corresponding to the x(v)’s for v > t must be zero.

This then requires the impulse response of a causal continuous time LTI system satisfy the following conditions :

 

 As stated before in the discrete time analysis,the system output depends only on the past and the present values of the input.

 Proof : ( By contradiction )

 Suppose, there exists > 0, such that h(-) 0.

  Now consider

Since,

         

   System is not causal, a contradiction. Hence,