Definition of a Constant Function
Definition
A function $c : X \to Y$ is called a Constant Function if it satisfies the following for all $x_{1} , x_{2} \in X$. $$ c \left( x_{1} \right) = c \left( x_{2} \right) $$
Explanation
Typically, the starting point where one first ‘recognizes’ a constant function as a function is when learning about the differentiation of constant functions. $$ \lim_{h \to 0} {{ c \left( x + h \right) - c \left( x \right) } \over { h }} = 0 $$ Up to that point in the curriculum, students often find it hard to understand what a function is, what numbers are, and if they are not top students, they may even absurdly categorize terms into ’letters’ and ’numbers’ (the author included). However, by differentiating both sides, one starts to ponder how to handle the part that is not a letter―not a polynomial function. Soon after dealing with indefinite integrals, $$ \int f(x) dx = F(x) + c $$ one becomes accustomed to the concept of constants by denoting ‘some constant $c$’. Interestingly, even jokingly, there comes a time when ‘constant functions’, which are not considered important in mathematics, universally appear in some field.
Continuity
If $X, Y$ is a topological space, one can discuss the continuity of a function. A constant function is trivially continuous in any space, and usually, continuous functions like $f : X \to \mathbb{Z}$ appear in arguments stating that ‘because the function values that are integers cannot change continuously, $f$ is none other than a constant function’.