Analysis I | Part IA, 2001

Suppose that ff is a continuous real-valued function on [a,b][a, b] with f(a)<f(b)f(a)<f(b). If f(a)<v<f(b)f(a)<v<f(b) show that there exists cc with a<c<ba<c<b and f(c)=vf(c)=v.

Deduce that if ff is a continuous function from the closed bounded interval [a,b][a, b] to itself, there exists at least one fixed point, i.e., a number dd belonging to [a,b][a, b] with f(d)=df(d)=d. Does this fixed point property remain true if ff is a continuous function defined (i) on the open interval (a,b)(a, b) and (ii) on R\mathbb{R} ? Justify your answers.

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