Question

Water can exist remain same in water and ice cubes at 0° celcius

Answer 1

Water exists in three distinct phases at something called the triple point. Zero degrees celsius is defined by the triple point of water which is 273.16K at 611.2 Pa. 

At this temperature water is in the process of changing from a solid state into the liquid phase or visa versa. Molecules in the liquid phase can loose a bit of energy and solidify whilst solid water (ice) can gain some energy and melt. This can be seen in melting ice where the solid ice exists for some time while the exposed surface melts.