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# Continuity

If there are no holes in your function, it's continuous! Many powerful theorems in Calculus only apply to these special types of functions.

The function \(f,\) some of whose values are shown below, is a continuous function defined for all real numbers.

\[ \begin{array} {c|c|c|c|c|c|c|c} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 10 & 17 & 8 & -11 & -10 \\ \end{array} \]

Between which two \(x-\)values does the intermediate value theorem guarantee a solution to \(f(x) = 0?\)

True or False? There is a maximum altitude in the set of altitudes reached by Dylan during the interval \([0,5].\)

True or False: Mehmet must have visited at least one place that has the same altitude as his home.

True or False: Mehmet must have visited at least one place outside of his local apple vendor's neighborhood where apples are the same price as they are at home.

On which interval(s) does the function \(f(x) = x^2\) attain a maximum value?

\(A = [1,2] = \{x | 1 \leq x \leq 2 \} \,\,\,\,\,\,\)

\(B = [1,2) = \{x | 1 \leq x < 2 \} \,\,\,\,\,\,\)

(The number 2 is in \(A\) but not in \(B.)\)

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