MA005: Calculus I

Since the graph of a continuous function is connected and does not have any holes or breaks in it, the values of the function can not "skip" or "jump over" a horizontal line (Fig. 8). If one value of the continuous function is below the line and another value of the function is above the line, then somewhere the graph will cross the line. The next theorem makes this statement more precise. The result seems obvious, but its proof is technically difficult and is not given here.

If the graph of $\mathrm{f}$ connects the points $(\mathrm{a}, \mathrm{f}(\mathrm{a}))$ and $(\mathrm{b}, \mathrm{f}(\mathrm{b}))$ and $\mathrm{V}$ is any number between $\mathrm{f}(\mathrm{a})$ and $\mathrm{f}(\mathrm{b})$, then the graph of $\mathrm{f}$ must cross the horizontal line $\mathrm{y}=\mathrm{V}$ somewhere between $\mathrm{x}=\mathbf{a}$ and $\mathbf{x}=\mathbf{b}$ (Fig.9). Since $\mathrm{f}$ is continuous, its graph cannot "hop" over the line $\mathrm{y}=\mathrm{V}$.

Most people take this theorem for granted in some common situations:

• If a child's temperature rose from $98.6^{\circ}$ to $101.3^{\circ}$, then there was an instant when the child's temperature was exactly $100^{\circ}$. In fact, every temperature between $98.6^{\circ}$ and $101.3^{\circ}$ occurred at some instant.
• If you dove to pick up a shell 25 feet below the surface of a lagoon, then at some instant in time you were 17 feet below the surface. (Actually, you want to be at 17 feet twice. Why?)
• If you started driving from a stop (velocity $=0$) and accelerated to a velocity of 30 kilometers per hour, then there was an instant when your velocity was exactly 10 kilometers per hour.

But we cannot apply the Intermediate Value Theorem if the function is not continuous:

• In 1987 it cost $22¢$ to mail a letter first class inside the United States, and in 1990 it cost $25¢$ to mail the same letter, but we cannot conclude that there was a time when it cost $23¢$ or $24¢$ to send the letter. Postal rates did not increase in a continuous fashion. They jumped directly from $22¢$ to $25¢$.
• Prices, taxes, and rates of pay change in jumps, discrete steps, without taking on the intermediate values.

The Intermediate Value Property can help us finds roots of functions and solve equations. If $\mathrm{f}$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs (one is positive and one is negative), then $\mathrm{0}$  is an intermediate value between $\mathrm{f}(\mathrm{a})$ and $\mathrm{f}(\mathrm{b})$ so $\mathrm{f}$ will have a root between $\mathrm{x}=\mathrm{a}$ and $\mathrm{x}=\mathrm{b}$.