  We derive a fine-grained uncertainty relation for the measurement of two
incompatible observables on a single quantum system of continuous variables,
and show that continuous variable systems are more uncertain than discrete
variable systems. Using the derived fine-grained uncertainty relation, we
formulate stronger steering criterion that is able to reveal the steerability
of N00N states that has hitherto not been possible using other criteria. We
further obtain a monogamy relation for our steering inequality which leads to
an, in principle, improved lower bound on the secret key rate of a one-sided
device independent quantum key distribution protocol for continuous variables.
