Reducing the waist of an optical fiber taper to diameters below 1 microm can be interpreted as creating an optical nanofiber with propagation properties different from conventional optical fibers. Although there is theoretically no cutoff of the fundamental mode expected, a steep decline in transmission can be observed when the fiber diameter is reduced below a specific threshold diameter. A simple estimation of this threshold diameter applicable to arbitrary taper profiles and based on the diameter variation allowing adiabatic transmission behavior is introduced and experimentally verified. In addition, this threshold behavior is supported by investigating the variation of the power distribution of the nanofiber fundamental mode as a function of the fiber diameter.