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 $$f(c)=k$$.

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:

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 $$[0,8]$$.

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

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

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 $$f (x) =2x$$, 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 $$[0,1]$$.