Funções Injetivas e Monotonas

If $f$ is an injection in the range $[a,b]$ this means that for $x,y \in [a,b]$, $f(x) = f(y) \iff x=y$.

Let's think about this intuitively. Suppose on the interval $[a,b]$ the image of $f$ is increasing. If it's not monotone, then at some point later, the image of $f$ must be strictly decreasing. And the moment it starts decreasing we must have hit a value that we hit before.

Notice that the moment the image of $f$ starts to decrease above, we immediately start hitting multiple values.

We can continue to think about this intuitively. If the graph starts out increasing and then starts to decrease at some point, the point where it starts to decrease must be a local maximum. And on both sides of a local maximum, we must have points which are "less" than it. Since $f$ is continuous, we can't skip any values per se. Therefore, there must be some overlap and $f$ cannot be an injection. Maybe we can make a solution out of this!

Actually, formulating a rigorous proof around this idea turns out to be pretty hard. The reason for this is because there is no good way to talk about a function "increasing" or "decreasing" at a point.

Intuitively, though, hopefully this question makes sense. Now let's think of other ways we can approach this question. If you haven't already, take a moment to think about how we could approach this question, whether it be by formulating the statement differently or attacking it from a different angle.

The first thing to note is that we definitely want to invoke Intermediate Value Theorem somehow. Since IVT tells us that the image of $f$ must achieve a certain value, we probably want to prove this question using contradiction.

So let's assume for the sake of contradiction that we have a continuous injection that is not monotone.

We want to somehow translate this into math that makes it easy to incorporate IVT. How can we do this? Remember, IVT tells us that if the image of $f$ on some interval ranges from $[a,b]$ then every value $a \le x \le b$ in between must be achieved in the image.

Choose arbitrary $c,d \in [a,b]$ such that $c < d$