Finding a function based on their limits I have points and limits of a function and even the shape of the function and I'm looking for the function, something that very interesting for me how could I control the curve of the function?

(1) $\lim\limits_{x \to inf} f(x) = 1 $

(2) $f(\frac{1}{c}) = 1 $

(3) $0\lt x$

(4) $0\lt c \leq 1$
One such function is $1-(1-c^{2}x^{2})^{+}$ where $y^{+}$ denotes $\max \{y,0 \}$. 

Edit: this function is convex on $(0,\frac 1 c)$. To make it concave consider $\min \{1, \frac {1-e^{-cx}} {1-e^{-1}} \}$.
This function satisfies all conditions except one of them, but maybe it'll help you, try $f(x)= -\frac{1}{x+1} +1$