Timeless Theorems of Mathematics/Intermediate Value Theorem

The Intermediate Value Theorem is a fundamental theorem in calculus. The theorem states that if a function, $f(x)$ is continuous on a closed interval $[a,b],$ then for any value, $y$ defined between $(f(a)$ and $f(b),$ there exists at least one value $c\in (a,b)$ such that $f(c)=y$.

Intermediate Value Theorem: f(x) is continuous on [a, b], there exists at least one value c, that is defined on (a, b) such that f(c) = y.

Proof

Statement: If a function, $f$ is continuous on $[a,b],$ then for every $y$ between $f(a)$ and $f(b),$ there exists at least one value $c\in (a,b)$ such that $f(c)=y$

Proof: Assume that $f(x)$ is a continuous function on $[a,b]$ and $f(a)<f(b).$

Consider a function $g(x)=f(x)-y.$ The purpose of defining $g(x)$ is to investigate the behavior of $f(x)$ concerning the value $y$.

Since $f$ is continuous on $[a,b]$ and $y$ is a constant, $g(x)=f(x)-y$ is also continuous on $[a,b],$ as the difference of two continuous functions is continuous.

Now, $f(a)<y$ [As $c\in (a,b)$ and $y=f(c)$]

Or, $f(a)-y<0$

$\therefore g(a)<0$

In the same way, $g(b)>0$

Since $g(x)$ is continuous and $g(a)$ is defined below the $x$-axis while $g(b)$ is defined above the $x$-axis, there must exist at least one point $c$ in the interval $[a,b]$ where $g(c)=0$.

Therefore, at the point $c$, $g(c)=f(c)-y=0\implies f(c)=y$

∴ There exists at least one point $c$ in the interval $[a,b]$ such that $f(c)=y.$ [Proved]