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 $$f(c) = y$$.

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 $$f(a) 0$$

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 $$c$$, $$g(c) = f(c) - y = 0 \implies f(c) = y$$

&there4; There exists at least one point $$c$$ in the interval $$[a,b]$$ such that $$f(c)=y.$$ [Proved]