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Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties.

  1. $f(0)=0$,
  2. $f(1)=1$, and
  3. $f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_2<1$.

Then the number of such functions is

  1. $0$
  2. $1$
  3. $2$
  4. $\infty$
in Calculus
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all series of polynomial functions? like x^2 , x^3 ?

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