Continuity

# other theorems

**Bolzano-Weierstrass’s Theorem:** If *f* is continuous in *[a,b]*, then *f* has at least one absolute maximum and one absolute minimum in the interval. That is

**Intermediate-value Theorem:** If *f* is continuous in *[a,b]* and

Demo. let *g(x) = f(x) – k* →*g* is continuous in *a*, and

then

(Bolzano's Theorem)

In every case

**Exercise**: Demonstrate that exists a value *c* in (2,5) such that *f(c)* = 20/3, if

Solution: Use the Intermediate-value Theorem in this interval.

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