The intermediate value theorem states the following: Consider an interval I = [ a , b ] in the real numbers ℝ and a continuous function f : I → ℝ . Then, If u is a number between f ( a ) and f ( b ), f ( a ) &lt; u &lt; f ( b ) (or f ( a ) &gt; u &gt; f ( b ) ),then there is a c ∈ ( a , b ) such that f ( c ) = u.What happens if we consider R instead of I ?Is there a simple method to prove the theorem in this case ? I know the method for f : I → R with I = [ a , b ], but here it is different.

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The intermediate value theorem states the following: Consider an interval   I  =  [  a  ,  b  ] in the real numbers       ℝ   and a continuous function   f  :  I  →      ℝ  . Then, If   u is a number between   f  (  a  ) and   f  (  b  ),   f  (  a  )  &amp;lt;  u  &amp;lt;  f  (  b  ) (or   f  (  a  )  &amp;gt;  u  &amp;gt;  f  (  b  ) ),then there is a   c  ∈  (  a  ,  b  ) such that   f  (  c  )  =  u.What happens if we consider       R   instead of   I ?Is there a simple method to prove the theorem in this case ? I know the method for   f  :  I  →      R   with   I  =  [  a  ,  b  ], but here it is different.

hitturn35 · Accepted Answer

The general equation for the point-slope form of a linear equation is       y    -          y      1        =    m          (      x      -              x        1            )      , where m is the slope, and             (              x        1            ,              y        1            )       is the known point.      y    -          (      -      5      )        =    -    5          (      x      -              (        -        3        )            )       =      y    +    5    =    -    5          (      x      +      3      )

The intermediate value theorem states the following: Consider an interval I=[a,b] in the real numbers R and a continuous function f:I->R. Then, If u is a number between f(a) and f(b), f(a)<u<f(b) (or f(a)>u>f(b) ), then there is a c∈(a,b) such that f(c)=u.

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