Image of Interval by Continuous Function is Interval

Theorem

Let $I$ be a real interval.

Let $f: I \to \R$ be a continuous real function.

Then the image of $f$ is a real interval.

Proof 1

Let $J$ be the image of $f$.

By definition of real interval, it suffices to show that:

$\forall y_1, y_2 \in J: \forall \lambda \in \R: y_1 \le \lambda \le y_2 \implies \lambda \in J$

So suppose $y_1, y_2 \in J$, and suppose $\lambda \in \R$ is such that $y_1 \le \lambda \le y_2$.

Consider these subsets of $I$:

$S = \set {x \in I: \map f x \le \lambda}$

$T = \set {x \in I: \map f x \ge \lambda}$

As $y_1 \in S$ and $y_2 \in T$, it follows that $S$ and $T$ are both non-empty.

Also, $I = S \cup T$.

So from Interval Divided into Subsets, a point in one subset is at zero distance from the other.

So, suppose that $s \in S$ is at zero distance from $T$.

From Limit of Sequence to Zero Distance Point, we can find a sequence $\sequence {t_n}$ in $T$ such that $\ds \lim_{n \mathop \to \infty} t_n = s$.

Since $f$ is continuous on $I$, it follows from Limit of Image of Sequence that:

$\ds \lim_{n \mathop \to \infty} \map f {t_n} = \map f s$

But:

$\forall n \in \N_{> 0}: \map f {t_n} \ge \lambda$

Therefore by Lower and Upper Bounds for Sequences:

$\map f s \ge \lambda$

We already have that:

$\map f s \le \lambda$

Therefore:

$\map f s = \lambda$

and so:

$\lambda \in J$

A similar argument applies if a point of $T$ is at zero distance from $S$.

$\blacksquare$

Proof 2

Let $J$ be the image of $f$.

By Subset of Real Numbers is Interval iff Connected we need to show that $J$ is connected (and hence an interval).

Aiming for a contradiction, suppose not.

Then there exists a separation $S \mid T$ of $J$.

Define $A = f^{-1} \sqbrk S$ and $B = f^{-1} \sqbrk T$. $A$ and $B$ are both non-empty.

Because $f$ is continuous, by Continuous iff inverse image of any open set is open we must have $A$ and $B$ open.

Because $S \mid T$ is a separation:

$A \cap B = f^{-1} \sqbrk S \cap f^{-1} \sqbrk T = f^{-1} \sqbrk {S \cap T} = \O$, because $S \mid T$ is a separation.

Also, $A \cup B = f^{-1} \sqbrk S \cup f^{-1} \sqbrk T = f^{-1} \sqbrk {S \cup T} = f^{-1}(J) = I$ ($S \mid T$ is a separation of $J$).

Hence $A \mid B$ is a separation of $I$.

$I$ can certainly not be an interval (because it is not connected).

This is a contradiction.

Thus $J$ must be an interval.

$\blacksquare$