Circuit with adjustable source and a diode
If you searched for a typical voltage versus current graph for any number of silicon diodes you would find something like this: -



As you can see, the diode is forward conducting at certainly 0.7 volts but, doesn't appear to be conducting if the forward voltage reduces to maybe 0.4 volts. 

That's what this simplistic graph tells you but it isn't true. Take a look at a logarithmic graph of forward voltage and forward current: -



It's a striaght line and you can see that every tenfold reduction in current is brought about by a forward volt drop reduction of about 100 mV. As current eventually falls towards zero that relationship starts to become a linear relationship with 0 V and 0 nA being truly representative at the origin of the graph.

This means that a diode is always conducting unless the applied voltage is exactly zero.

Given that the circuit in the question has a more positive bias voltage at the anode compared to the cathode, whenever there is a positive voltage produced by the source, there will be forward conduction of some current.

If you were to ask a question that said what voltage would be the source when say 1 mA flowed or maybe when the forward voltage became 0.7 volts that would be different.