MA101.6 Monotonic Functions

A function f is said to be monotonic increasing in (a,b) if \forall x_1,x_2 in this interval, x_1 <  x_2 \Rightarrow f(x_1) \leq f(x_2).

The function is said to be strictly monotonic increasing in (a,b) if \forall x_1,x_2 in this interval, x_1 < x_2 \Rightarrow f(x_1) < f(x_2).

A function f is said to be monotonic decreasing in (a,b) if \forall x_1,x_2 in this interval, x_1 < x_2 \Rightarrow f(x_1) \geq f(x_2).

The function is said to be strictly monotonic decreasing in (a,b) if \forall x_1,x_2 in this interval, x_1 < x_2 \Rightarrow f(x_1) > f(x_2).

Example

Let f be a strictly monotonic increasing function. Prove that f is an injection.

Solution

Let f be a strictly monotonic increasing function.
Let x_1,x_2 \in D(f) be such that f(x_1) = f(x_2).
Now x_1 < x_2 \Rightarrow f(x_1) < f(x_2) which is a contradiction, and x_2 < x_1 \Rightarrow f(x_2) < f(x_1) which also is a contradiction.
So x_1 = x_2 and thus f is an injection.

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