I am trying to understand and prove the fundamental theorem of calculus and I ran into some confusion understanding the intermediate value theorem . several sources online claim that if a function f(x) is continuous on [a,b] let s be a number such that f ( a ) &lt; s &lt; f ( b ) then there exists a number k in the open interval (a,b) such that f(k)=s my question is why do we only assume the open interval shouldn't it also include the closed interval [a,b] and also why does s have to be less than both f ( a ) and f ( b )?

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I am trying to understand and prove the fundamental theorem of calculus and I ran into some confusion understanding the intermediate value theorem . several sources online claim that if a function f(x) is continuous on [a,b] let s be a number such that   f  (  a  )  &amp;lt;  s  &amp;lt;  f  (  b  ) then there exists a number k in the open interval (a,b) such that f(k)=s my question is why do we only assume the open interval shouldn&#039;t it also include the closed interval [a,b] and also why does s have to be less than both   f  (  a  ) and   f  (  b  )?

Allyson Gonzalez · Accepted Answer

The assertion “there is some   k  ∈  (  a  ,  b  ) such that   f  (  k  )  =  s” is stronger than the assertion “there is some   k  ∈  [  a  ,  b  ] such that   f  (  k  )  =  s”. So, why would we state a weaker statement when we can as easiy prove a stronger one.And, if you have in mind the statement “if   f  (  a  )  ⩽  s  ⩽  f  (  b  ), then there is some   k  ∈  [  a  ,  b  ] such that   f  (  k  )  =  x”, then that statement is trivial if   s  =  f  (  a  ) (just take   k  =  a then) or if   s  =  f  (  b  ) (just take   k  =  b then). So, the non-trivial part is when   f  (  a  )  &amp;lt;  s  &amp;lt;  f  (  b  ).

I am trying to understand and prove the fundamental theorem of calculus and I ran into some confusio

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