Mathematically speaking, is a "signal" a function or the set of outputs from a function In engineering it is well known that we abuse the notation of a function $f: A \to B$ with the image of that function $f(t) = \left\{y \in B\mid f(t) = y, t \in A\right\}$


When we say: let $f(t) = \sin(\omega t)$ be a signal, do we mean a function or the set of output from that function i.e. is signal a function or the output?
Similarly, looking at the diagram below, would one say that $\epsilon(t)$ is a set of inputs (i.e. scalars) going into the controller, or a function?
A signal is a physical quantity (e.g. voltage) carrying information, or a set of values (e.g. samples in discrete case) of the given function for different values of the underlying independent variable. 

In the diagram above $\epsilon(t)$ is a mathematical modeling (i.e. function) that characterizes the set of inputs (i.e. scalers) going into the controller. 

Strictly speaking, a signal is the "observed" (measurable) physical quantity while a function is a mathematical model (abstract representation) describing the relationship between the involved variables.