There is no material that shows all of these properties at the same time. Very hard materials, have lower toughness and break more easily. The following cutting tool materials are used:
Tool steels: They are relatively cheap and tough. Their hardness is sufficient to machine other steels.
Carbon tool steels: They lose their hardness at 200°C
High speed steels: They lose their hardness at 600°C, and are widely used in machining. Due to their ability to retain hardness at higher temperature, higher cutting speeds are possible.
Cutting ceramic: They are even harder than cemented carbides but have lower toughness. Aluminium oxide and silicon nitride are used. The latter has higher toughness, but can't be used for machining steel, due to very high wear. Alumina, or aluminum oxide, is also extensively used in the production of cutting tools due to its exceptional hardness and wear resistance. Alumina cutting tools also have high thermal conductivity, which helps dissipate heat generated during cutting, minimizing thermal damage to both the tool and the workpiece. Additionally, alumina's chemical stability ensures that it does not react with materials being cut, preventing contamination and maintaining the purity of the finished product.
↑Kumar, A.S.; Durai, A.R. (2003). "Machinability of hardened steel using alumina based ceramic cutting tools". International Journal of Refractory Metals and Hard Materials. 21 (3–4): 109–117. doi:10.1016/S0263-4368(03)00004-0.