Why is the intermediate value theorem so important? I would like to know why the intermediate value theorem is so important. So my questions are:

  
*
  
*Which important theorems do we prove using the intermediate value theorem?
  
*Are there direct applications of the intermediate value theorem outside mathematics?
  
*Does the intermediate value theorem have a historically importance?
  

 A: We can use the intermediate value theorem to compute equations like, for example, $\cos(x)=x$.
consider the function $f:[0,\pi/2]\mapsto\Bbb R, x\to\cos x-x$.
this function is continuous and $f(0)=1,f(\pi/2)=-\pi/2$. by the intermediate value theorem we know that $f(c)=0,c\in[0,\pi/2]$. this doesn't help us too much, so let's divide $[0,\pi/2]$ into $[0,\pi/4],[\pi/4,\pi/2]$.
now we have $f(0)=1,f(\pi/4)\approx -0.0782913822,f(\pi/2)=-\pi/2$.
using the intermediate value theorem we know that $c\in[0,\pi/4]$, great! now let's divide our interval into 2 smaller intervals again:$[0,\pi/4]$ into $[0,\pi/8],[\pi/8,\pi/4]$
now we have $f(0)=1,f(\pi/8)\approx0.531180451,f(\pi/4)\approx -0.0782913822$.
hence $c\in[\pi/8,\pi/4]$. by doing this over and over again we can get to pretty nice approximation 

this theorem  is important in physics where you need to construct functions using results of equations that we know only how to approximate the answer, and not the exact value, a simple example is 2 bodies collide in $\mathbb R^2$. in this case you will have system of 2 equations in similar form to the example of the first part.

I know about only one historically importance, before there was a definition of continuity people use this theorem, become if this is true for all sub-intervals in the function $f$ then $f$ is continuous
A: A bit too long for a comment ... 
In general, the IVT states that continuous maps preserve connectedness.
For point (1): 
Let $f$ be a real-valued continuous function defined on an interval $J$ of $\mathbb R$.  Claim: If $f$ is injective, then it is strictly monotone.
Define $g\colon D:=\{(x,y)\in J^2|x>y\}\to\mathbb R$, $g(x,y):=f(x)-f(y)$. As $f$ is continuous so is $g$.  Now $D$ is connected, hence $g(D)$ is connected, that is, an interval of the reals.  Since $f$ is injective, $g(D)$ doesn’t contain $0$.
A: there exist two antipodal points on the equator that have the same temperature.
Can you prove that at all times there is a pair of opposite points on the equator where the temperature at one is equal to the temperature of the other? 
To see how we can prove this, pick two arbitrary points. Label the points A and B with the temperature of A greater than B. Find the difference in temperature of the two points. If the difference is zero, then you've found a pair of points. If not, move point A  to point B and point B to point A. Now the difference in temperature is negative. Because the difference in temperature is a continuous function, there exists a point on the equator where the difference is zero. Those two points have the same temperature. 
