Quick True and False Calc Question-Please Help Quickly?

True or False: If f(x) is continuous and 0 %26lt;=f(x)%26lt;=1 for all x in the interval [0,1], then for some number x, f(x) = x. Explain your answer.





Please tell me whether true or false and have a brief explanation.





Thanks.Quick True and False Calc Question-Please Help Quickly?
TRUE!


if the function is continuous and always takes values between 0 and 1 on the interval [0,1] then there must be a number in there somewhere where f(x) = x


If you draw any function where this is true and the function f(x) = x then the curves must meet at some pointQuick True and False Calc Question-Please Help Quickly?
Mean value theorem says true.





If you draw a line along y=x.





If the value of f(x) is between 0 and 1, you can not create a continuous function from x=0 to x=1 without crossing this line.





Note: It is easy to create such a function on the intervals


(0,1], [0,1) and (0,1).