By: Michael Mo

The Intermediate Value Theorem – IVT – is a concept in calculus that talks about the property of a continuous function. Not that continuity on its own has a rigorous definition based on the concept of limits. For the sake of simplicity, however, we can just assume that a continuous function is one that you can draw without lifting your pen from the graph paper. With that in mind, IVT states that if a function is continuous over the interval [a, b] and a value N (the intermediate value) lies between f(a) and f(b), then there MUST BE a value c in between a and b such that f(c) = N.

As a heads up for those who plan to take calculus in the future, every other concept in calculus will appear equally confusing as the IVT. Nevertheless, I will soon show that IVT is actually not as brain-degrading as it seems.

## Example Easy: Growth Spurt

When I began puberty 4 years ago, my height was 150cm. Now, my height has continuously increased to 165cm (still short, but that is irrelevant). IVT essentially states is that I was at a height between 150cm and 165cm (N) some time (c) during this 4-year period. Knowing that my height was once 160cm, IVT confirms that I *must be *that tall sometime between 2017 and 2021.

## Example Practical: Temperature

With IVT, we can prove that there must be a pair of opposite points on earth that has the same temperature at any given time.

First and foremost, we have to recognize that the temperature around the globe changes continuously without any sudden jumps. For this reason, if we start rotating the opposite points and taking their differences at the new opposites, this function for temperature differences will also be continuous.

If we assume that Red T1 – Blue T1 yields a *positive difference*, then Red T – Blue T will yield a *negative difference* when we rotate those points exactly 180 degrees around. Since the change in temperature is continuous, there must be some opposite points whose difference equals to zero (*zero is the intermediate value*). **This precisely means that the temperature at those opposite points is the same**.

## Example Math: Function

If we have the equation:

We can use IVT to prove that there *must* exist a positive solution between two consecutive integers. First, we could multiply both sides of the equation by x to get:

Then, we let

f(x) = 0 is a solution to our original equation.

For now, I am going to simply state that this polynomial function is continuous everywhere (the proof would be a different topic). Therefore, we can let f(x) = 0 be the intermediate value N.

Notice that:

Since -1 < N < 1, IVT tells us that there must be a value x between 0 and 1 that satisfy f(x) = N. In other words, we have just shown that a solution between 0 and 1 (hence positive) exists for the original equation.

## Ideas to Consider

Perhaps your acute sense of mathematics has noticed IVT will not give you any *type* of numerical answer to a question. That is completely true; IVT cannot determine *when* something will happen, but simply *if* something will happen or not. In fact, we might never find a precise answer to a question even if IVT tells us that an answer exists. On a deeper thought, life is all about IVT as well; you can just never find out *what* and *when* something will change your life forever.

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