# Remark the theorem remains valid if f a d f b the

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Remark. The theorem remains valid if f ( a ) > d > f ( b ) (the simplest proof is obtained by applying the previous result to the function g ( x ) = - f ( x )). This (formally different) result is also called the intermediate value theorem. 2

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Corollary. If f : [ a, b ] R is continuous function on the closed bounded interval [ a, b ] then its range f ([ a, b ]) is the closed interval [ m, M ] where m and M are the g.l.b. and l.u.b. of f . Proof. By the maximum and minimum theorems, the function f attains its g.l.b. and l.u.b. at some points x 1 [ a, b ] and x 2 [ a, b ]. Applying the intermediate value theorem to the interval [ x 1 , x 2 ], we see that f takes all possible values between m and M . Example. The equation ( x 2 + 1) sin x = 1 has a solution in the interval [0 , π/ 2]. Indeed, ( x 2 + 1)(sin x ) - 1 = - 1 at the point x = 0 and ( x 2 + 1)(sin x ) - 1 = π 2 / 4 at the point x = π 2 . Since the function ( x 2 + 1)(sin x ) - 1 is continuous, it takes all intermediate values (including 0) on the interval [0 , π/ 2]. We know that sums, products and quotients of continuous functions are continuous functions. It turns out that the composition of two continuous functions is also continuous. Theorem. Let f be a continuous function on a closed interval [ a, b ], and let [ m, M ] be its range. If g is a continuous function on [ m, M ] then the composition g f ( x ) = g ( f ( x )) is a continuous function on [ a, b ]. Proof. Let c [ a, b ] and x n [ a, b ] be an arbitrary sequence which converges to c . Then, since f is continuous, f ( x n ) converges to f ( c ) [ m, M ]. Now, since g is continuous, g ( f ( x n )) converges to g ( f ( c )). Thus we have g f ( x n ) g
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