Intermediate Value Theorem (Darboux's Property)

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 $[0,1]$.