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.
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?
As Gilles says in his answer, when we talk about signals in signal processing, we generally mean the actual voltage / current / measurement values rather than the function (i.e. specific parts of the image of the function).
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?
For the feedback system then, in this case, I would say that we want both: we want to be able to measure the error $\epsilon(t)$ at a particular time $t$ and we want to be able to manipulate the functions in the diagram to be able to find the relationship between $Y(t)$ and $R(t)$ (and everywhere in between).
