1. The Intermediate Value Theorem (IVT) states that if a function $f$ is continuous on a closed interval $[a,b]$, and $d$ is any number between $f(a)$ and $f(b)$, then there exists at least one $c$ in $[a,b]$ such that $f(c) = d$. 2. The IVT has two key parts to understand: a. Continuity: The function must be continuous on the entire interval $[a,b]$. This means there are no breaks, jumps, or holes in the graph of the function. b. Intermediate Value: For any value $d$ between $f(a)$ and $f(b)$ (including $f(a)$ and $f(b)$ themselves), the function attains the value $d$ at some point $c$ in the interval. 3. The difference between the two parts is: - The first part is a requirement about the nature of the function (continuity). - The second part is the guarantee or conclusion about the existence of a point $c$ for any intermediate value $d$. 4. In simpler terms, continuity ensures no gaps, and the theorem guarantees that all values between the start and end outputs are hit by the function somewhere in the interval. 5. Example: If $f(1) = 3$ and $f(4) = 7$ and $f$ is continuous on $[1,4]$, then for any $d$ in $[3,7]$, there exists $c$ in $[1,4]$ such that $f(c)=d$. For instance, if $d=5$, then there is some $c \in [1,4]$ with $f(c) = 5$.