The correct answer is: A. Linear and time-variant.
A linear system is a system that satisfies the superposition principle. This means that the output of the system is the sum of the outputs that would be produced by each input signal individually. A time-variant system is a system whose output depends on the time at which the input is applied.
In the given system, the output $y(t)$ is equal to the integral of the input $x(u)$ over the interval $[t-T,t]$. This means that the output is a linear combination of the input signals, and the system is therefore linear. The output also depends on the time $t$, so the system is time-variant.
The other options are incorrect. Option B is incorrect because the system is not time-invariant. Option C is incorrect because the system is linear. Option D is incorrect because the system is time-variant.