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.