Formulating a function.Trying to understand normal distribution.

Question

How would a person go about formulating the normal distribution function satisfying the following conditions, for the first time ever?
The conditions are

1)The function must be defined for all real number

2)The function's output value at any point must lie between zero and 1 i.e. (0,1)

3)The total area under the graph of the function must be equal to one.

4)The output value decreases as we move away from a point 'A' in the input.

Please give step by step instructions if possible. I am trying to understand how normal distribution function was formulated from these conditions.