Its not the the sword would be unbreakable and unbendable, its more something in the likes..
If the metal is too soft, the sword bents so it does not break. But it loses its sharpness really easily
If the metal is too hard, the sword is razor sharp, but it snaps easily.
So japanese swords, are made with multiple metal layers. In simple terms, the core is a soft'er metal so the blade does not snap (so easily), and the edge is made of a hard'er metal so it keeps its edge. The curvature of those blades can be attributed to a natural difference when cooling, due to the fact that are 2 different types of metal. Its pretty a clever way of dealing with tecnological limitation.
That said, nowadays we have spring steel, which is basicaly both. Keeps edge for a good time, and bents instead of snapping.