Proving continuity using the IVT (1 Viewer)

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these are questions from Calculus by spivak 3rd edition.

7) How many continuous functions f are there which satisfy (f(x))^2= x^2 for all x?

8) Suppose that f and g are continuous, and that f^2 = g^2, and that f(x) ≠ 0 for all x. Prove that either f(x) = g(x) for all x, or else f(x) = -g(x) for all x.

10) Suppose f and g are continuous on [a, b] and that f(a) < g(a), but f(b) > g(b). Prove that f(x) = g(x) for some x in [a, b]. (It is going to be a very short proof)

11) Suppose that f is a continuous function on [0, 1] and that f(x) is in [0, 1] for each x (draw a picture). Prove that f(x) = x for some number x.

i dont understand how to go about these problems
 
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Let's start with 7. What candidate functions do you have for f(x)?
 
i can think of the identity function f(x) = x
 

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