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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §2.7 Continuous Functions, #35 · Calculus · Continuity

(d) If … and … and … is continuous on …, then…

True or false, with an example to illustrate:

(a) If $f (x)$ is continuous at all $x$ , it has a maximum value $M$ .

(b) If $f (x) \leq 7$ for all $x$ , then $f$ reaches its maximum.

(c) If $f (1) = 1$ and $f (2) = - 2$ , then somewhere $f (x) = 0$ .

(d) If $f (1) = 1$ and $f (2) = - 2$ and $f$ is continuous on $[1, 2]$ , then somewhere on that interval $f (x) = 0$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §2.7 Continuous Functions, #35

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