Property of Darboux (theorem of the intermediate value)

Let $f (x)$ be a continuous function defined in the interval $[a, b]$ and let $k$ be a number between the values $f (a)$ and $f (b)$ (such that $f (a) \leq k \leq f (b)$ ).

Then some value $c$ exists in the interval $[a, b]$ such that $f (c) = k$ .

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This property is very similar to the Bolzano theorem. In fact it is possible to deduce it very easily:

Taking the function $g (x) = f (x) - k$ it is easy to see that it will satisfy the Bolzano theorem:

As $f (a) \leq k \leq f (b) \Rightarrow f (a) - k \leq 0 \leq f (b) - k \Rightarrow g (a) \leq 0 \leq g (b) \Rightarrow$

$\Rightarrow g (a) \cdot g (b) \leq 0$ , then by Bolzano a value $c$ exists in the interval $[a, b]$ such that $g (c) = 0$ .

But of course $0 = g (c) = f (c) - k \Rightarrow f (c) = k$ and the property of Darboux is proven.

Let's see some examples of application:

Example

We are going to look for the existence of a solution to the equation $(x - 1)^{3} = 2$ .

We define the function $f (x) = (x - 1)^{3}$ .

We have to look for an interval such that the value $2$ falls inside .

Let's take, for example, the interval $[1, 3]$ .

The image of the interval is $f ([1, 3]) = [f (1), f (3)] = [0, 8]$ and clearly the value $2$ belongs to it.

Therefore, we can be assured of the existence of at least one solution to the equation $(x - 1)^{3} = 2$ in interval $[0, 8]$ .

Example

We will look to see if solutions for the equation $3 = e^{x} + 2 x$ exist.

We define the function $f (x) = e^{x} + 2 x$ .

We have to look for an interval such that its image contains the value $3$ .

For example, we are going to evaluate the function in: $\begin{array}{rcl} f (0) & = & 1 \\ f (1) & = & e + 2 > 3 \end{array}$

Moreover, the exponential function is increasing, as is the function $f (x) = 2 x$ , so our function is increasing and consistently the image of $[0, 1]$ contains the value $3$ .

Therefore, using the property, we can be sure that at least one solution to our equation exists inside the interval $[0, 1]$ .