Why is $\equiv$ used for functions that are Identically Zero? I have encountered contexts in which the "definition" or "equivalent" sign "$\equiv"$ is used to make definitions of variables or functions. 
However, a common way authors use it is when a functions is identically zero. For example, if some function is determined to be "trivial," it is not unusual to see a statement like "...so then the function $f$ would be identically zero: $f \equiv 0$."
Why is it so common to use the $\equiv$ sign when an equals sign would seem to serve the same purpose above?
 A: The sign you are referring to is the standard symbol for "identically equal to". For example, a more usual context in which to see that symbol (at least in pre-calculus classes) might be: $$(x-1)^{2} \equiv x^{2}-2x+1,$$ where the presence of the symbol indicates that the equation holds for all values of $x$.
The reason for its use in this specific context is that the author wants to emphasize that the function is vanishing everywhere, i.e., identically vanishing, i.e., "identically equal to" $0$.
Note that many authors intentionally do not use $\equiv$ when they are defining things; I presume they maintain this practice so that questions like this don't come up. Instead, such authors might use $:=$, which has the advantage that it is meaningfully reversible, and many authors don't use a specific symbol and just use $=$, stating that the equation is meant to be taken as a definition in prose.
A: $x+x-2x \equiv 0$  but $x^2-1=0$ is for only $x= \pm 1$.  The first represents an identity which holds for all $x$ while the other is conditional equal which may or may not have solutions.
