Graph of Real Bijection in Coordinate Plane intersects Horizontal Line at One Point

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Theorem

Let $f: \R \to \R$ be a real function which is bijective.

Let its graph be embedded in the Cartesian plane $\mathcal C$:


Graph-of-bijection-intersect-horizontal.png


Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on exactly one point $P = \tuple {a, b}$ where $b = \map f a$.


Proof

By definition, a bijection is a mapping which is both an injection and a surjection.


Let $\mathcal L$ be a horizontal line through a point $b$ in the codomain of $f$.


From Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line:

$\mathcal L$ intersects the graph of $f$ on at least one point $P = \tuple {a, b}$ where $b = \map f a$.


From Graph of Real Injection in Coordinate Plane intersects Horizontal Line at most Once:

$\mathcal L$ intersects the graph of $f$ on at most one point $P = \tuple {a, b}$ where $b = \map f a$.


The result follows.

$\blacksquare$


Sources