Definition 1.1 (Function): A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain* of

Here are the definitions and remarks from the provided text:

• Definition 1.1 (Function): A function $f$ consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain of the function, while the set of outputs is known as the range of the function.
• Definition 1.2 (Function of a Real Variable): Let $N \subset \mathbb{R}$. A function $f$ of a real variable is a rule or law which assigns to each $x \in N$ exactly one $y \in \mathbb{R}$. The variable $x$ is called the argument or the independent variable, and the variable $y$ is termed the dependent variable.
• Definition 1.3 (Domain, Range and Co-domain): The set $N$ is known as the domain of function $f$ and is denoted by $D(f)$ or $\text{Dom}(f)$. Similarly, a set $\{y = f(x) : x \in D(f)\}$ is called the range of $f$, denoted by $R(f)$. The general term for function is called mapping. Range is a subset of the co-domain.
• Remark 1.1:
  1. The domain is a part of the definition of the function. If the domain is not specified, we take it as the natural domain, which is the largest possible domain.
  2. Two functions $f$ and $g$ are equal, that is, $f = g$, if the following two conditions are satisfied:
     • $D(f) = D(g)$ (they have the same domain).
     • $f(x) = g(x)$ for all $x$ in their common domain.