1. function continuity proof

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hello:
here is my problem:
if a function f(x)is continuous in [0,1],and its scope is also [0,1],then there must be one point e to make f(e)=e
if i use the theorem a continuous function can have any return value between upper and lower bound,then i can have
e belong to [0,1],and there will be a vulue a to make f(a)(a also belong to the same definition scope[0,1])to have a value e,but how can i promise the there can must be a chance in which a=e

any comment is appreciated.

2.

3. Are you sure you copied this correctly? The usual form would be;
if f is continuous in [0,1], then there is an a in [0,1] such that f(0)<f(a)<f(1), and you may set f(a) = e.

If you insist also that f(e) = e, then either f is trivial, or e = 0, And if f(a) = e = f(e), then a = e, but I can't beleive you stated the problem correctly

4. The probelm is fine as stated.

Hint: consider the function g(x)=f(x)-x. Apply intermediate value theorem.

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