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This statement is incorrect in its use of the word always. Higher frequency light only emits electrons with higher kinetic energies if the energy has already surpassed that of the work function; if the light's energy has not yet reached that of the work function, no electrons would be emitted.
That statement would only be true given that the frequency is high enough to eject the electron from the metal surface in the first place. For example, it could be possible that the increase in frequency (and thus energy) of the incident light is still less than the work function of the metal.
Wouldn't increasing the frequency of a light even if it originally did not have the energy to emit an electron eventually lead to an electron being emitted from the metal, because if E=hv and v increases, wouldn't the energy of the photon increase until it surpasses the work function?
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