Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line

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Theorem

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

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


Graph-of-surjection-intersect-horizontal.png


Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on at least 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 surjection.

Hence:

$\forall b \in \R: \exists a \in \R: b = \map f a$


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

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

The result follows.

$\blacksquare$


Sources