Intermediate Value Theorem (IVT)

If $f(x)$ is continuous on a closed interval $[a,b]$, then for every value $M$ strictly between $f(a)$ and $f(b)$, there exists at least one value $c$ such that $f(c)=M$.

Theorem 2: Initial Value theorem

If $f(t)$ is piecewise-continuous and ${\int }_0^{\infty }|f(t)|e^{-\alpha t}$ converges for some $\alpha \in \mathbb{R}$ then $f(0^{+})={\lim }_{s\to \infty }sF(s)$

Related

• Final Value Theorem