Single phase induction motor speed control (by VFD)

Many people keep telling that it's not possible to control the speed of those single phase induction motors that are found in lathes, compressors etc. by varying supply frequency. But why? I understand that they might not work very well (those motors are toys after all) if capacitors are used as capacitor impedance depends on frequency and they are desgined to run at 50/60Hz. But at least speed control should be possible to an extent, right? I have a 3-phase VFD device that I could use to test this but I'm not sure what happens if I connect a single phase load to it...

All induction motor have speed proportional with frequency. When reduce frequency, alson need to reduce voltage to keep same flux.
This is automaticaly made by inverter in various mode (depend of control method); when reduce voltage by k, torque is reduced by k^2, so torque may
became very low at small speed. May try to use 3-phase inverter for tests but check voltages and manual inverter also to find if there are some restriction /
protections to avoid single phase load. There are another single phase motor used in home-appliances, but with brushes; this motor is so called universal
motor and in fact is a DC motor with series excitation coil; this construction allow it to accept DC and AC voltage; for this motor type can't change speed by
changing frequency, as it is a DC motor (collector - brushes is an "mechanical rectifier"); is not recomanded to use this motor without load because speed
up very high and risk to destroy itself and lead to injuries.