# Memorylessness of simple delay system

As usual, $y(t)$ is the output signal of a system, and $x(t)$ is the input signal. I'm wondering whether or not a certain system has memory.

It's easy for me to see that the system

$y(t) = \int_{t-T}^{t} x(t) dt$ where $T > 0$

has memory, because an output at time $t_0$ depends on all values of input from $[t_0 - T, t_0]$.

But what about the following system, which just delays input by some constant $C$:

$y(t) = x(t - C)$ where $C > 0$

Upon first glance, this doen't feel like it has memory because there is no integral. But the output of a delay system does depend upon inputs from the past. Does that mean a simple constant delay system has memory?

A system is memoryless if its output ($y(t)$) for each value of the independent variable ($t$ in this case) at a given time is dependent only on the input at that same time ($x(t)$).
This can be seen just by replacing $t$ by some number. Take, for instance, $t=0$. In that case, the output for that instant ($y(0)$) depends on the input for another value of the independent variable ($x(C),C>0$); that means that the system has memory.