Fixed point
Elementary
Think of a function as a machine that changes its input into an output. A fixed point of the function is an input that the function leaves unchanged: The output equals the input.
High school
Consider a function $f$ whose inputs and outputs are the same kind of thing. A fixed point of $f$ is a point $x$ such that $f(x) = x$: the function leaves it unchanged.
Undergraduate
Consider a function $f : X \to X$. A fixed point of $f$ is a point $x \in X$ such that $f(x) = x$.
Examples
Examples
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$0$, for $x \mapsto x^2$
High school
$0$ is a fixed point of $x \mapsto x^2$, because $0^2 = 0$.
Non-examples
Things that may be mistaken for examples.
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$1$, for $\dot{x} = f(x) = 1 - x$
High school
$1$ is not a fixed point of $f$, because $f(1) = 0 \neq 1$.