When we say that a function u(t) has a weak singularity at t = 0, it means that the function exhibits some kind of irregular behavior or divergence as t approaches zero, but the singularity is not severe enough to cause the function to become infinite or undefined at that point.
A singularity, in general, refers to a point where a function becomes infinite, undefined, or exhibits abnormal behavior. It is a point of discontinuity or breakdown in the smoothness or regularity of the function. Singularity points can arise for various reasons, such as division by zero, taking the square root of a negative number, or when a function approaches a certain limit that it cannot reach.
A weak singularity is a type of singularity that does not result in an infinite or undefined value for the function at the singular point. Instead, the function may exhibit a jump, a sharp change, or a non-analytic behavior near the singularity, but it remains finite and well-defined. In other words, the function may have a finite value at the singular point but behaves differently from its behavior in the surrounding region.
To summarize, a singularity refers to a point where a function becomes infinite, undefined, or exhibits abnormal behavior, while a weak singularity is a type of singularity where the function shows some irregularity or divergence near the singularity, but it remains finite and well-behaved at that point.