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

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Theorem

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

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


Graph-of-injection-intersect-horizontal.png


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

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


Proof

From Equation of Horizontal Line, a horizontal line in $\mathcal C$ through the point $\tuple {0, b}$ on the $y$-axis has an equation $y = b$.


By hypothesis, $f$ is a injection.

Hence:

$\forall a_1, a_2 \in \Dom f: \map f {a_1} = \map f {a_2} \implies a_1 = a_2$

where $\Dom f$ denotes the domain of $f$.

Thus for each $b \in \Cdm f$ there exists exactly one ordered pair $\tuple {a, b}$ such that $b = \map f a$.

That is, there is exactly one point on $y = b$ which is also on the graph of $f$.

The result follows.

$\blacksquare$


Sources