Modern DMMs measure resistance by injecting a small current through the probes and measuring the voltage developed across the probe terminals (most sophisticated bench DMMs can perform more accurate 4-wire measurement, where the current is injected by a couple of probes which is different from the probes used for measuring the voltage drop).

In other words, ohm function makes the DMM a current source with constant current value. The so called *test current*, which is often specified in the DMM's manual. 

To allow measuring resistors in-circuit (which is a nice feature to have in a DMM) the test current is often made quite small, so that the internal current source can provide such current by generating a voltage small enough to avoid turning on semiconductor device junctions.

For example, if an ohm function range uses a 1uA test current, then it can be used to measure resistors with a value of about 0.5V/1uA = 500kohm without turning on silicon PN junctions (which turn on at about 0.6V).

Such a tiny current, however, when injected in a low resistance path will generate a tiny voltage, which may be difficult to measure by the internal DMM circuitry. E.g., the same 1uA current above will generate a scant 10uV voltage drop across a 10ohm resistor. 

Therefore, DMMs circuitry increase the test current when using ohm's ranges designed to measure low resistance values.

Since continuity testing is meant to detect such low resistance values, (assuming the continuity function uses the same circuit used for resistance measurements) the test current is usually quit substantial. 1mA is not uncommon. When the probes are applied to the circuit under test, if a high resistance path lies across the terminals, the current source is forced to generate a higher voltage in order to pump the test current in the circuit. This will turn on PN junctions, giving false reading for the resistance.

For example, with an 1mA test current, it is sufficient to have 500ohm  resistance across the probe terminals to generate a 500mV drop, which are enough to bring silicon PN junctions in conduction. This will clamp the voltage to that value, making the DMM show you a 500mV/1mA=500ohm value on the display.

If you want to check if this is the case, try and measure with another DMM the test current put out by your original DMM on continuity function. If it measures ~1mA you have a high chance that during the continuity measurement you turned on some silicon PN junctions.