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18 .
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: , [ a; b ]. (xn) xnk, x 0∈[ a; b ].
f [ a; b ], , , n ∈ N xn ∈[ a; b ], f (xn)> n. n 1,2,3,{\ldots}, (xn) [ a; b ], f (x 1)>1, f (x 2)>2, f (x 3)>3,..., f (xn)> n...
(xn) (xnk), x 0∈[ a; b ]: lim k →∞ xnk = x 0 (1)
(f (xnk)). f (xnk)> nk lim k →∞ f (xnk)=+∞ (2),
, (1) lim k →∞ f (xnk)= f (x 0) (3)
(2) (3) ( ). . . ...
1
, f [ a; b ], .
c =inf x ∈[ a; b ] f (x), d =sup x ∈[ a; b ] f (x),
.
2
, . , y = tgx, tgx ∈ C ((−2π;2π)), .
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, (.. ).
: f (x)∈ C ([ a; b ]), c =inf x ∈[ a; b ] f (x), d =sup x ∈[ a; b ] f (x).
c, d ∈ R. , f [ a; b ] , .. x 1, x 2∈[ a; b ], f (x 1)= c, f (x 2)= d.
, , x 2.
(∀ x ∈[ a; b ])(f (x)= d). , .. x 2, f (x 2)= d [ a; b ], [ a; b ] f (x)< d d − f (x)>0. ϕ(x)=1 d − f (x). ϕ(x) [ a; b ] ( [ a; b ] d − f (x)/=0), . ϕ(x) [ a; b ].
, >0 (∀ x ∈[ a; b ])(0<1 d − f (x)≤ M), f (x)≤ d −1 M < d.
, d f (x) [ a; b ], .. . x 2 , f (x 2)= d.
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x 1∈[ a; b ], f (x 1)= c.
f [ a; b ], [ a; b ] f , .. .
: [ a; b ] f [; d ], c =inf[ a; b ] f (x)=min[ a; b ] f (x), d =sup[ a; b ] f (x)=max[ a; b ] f (x), - ...
19 . .
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