Modern DMMs measure resistance by injecting a small current through the probes and measuring the voltage drop that develops across the probe terminals (most sophisticated bench DMMs can also perform more accurate 4-wire measurements, a.k.a. Kelvin measurements, where the current is injected by a pair of probes which is different from the probes used for measuring the voltage drop).
In other words, the ohm function makes the DMM behave like a current source with a constant current value. This is 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 quite small, so that the DMM's current source circuitry can provide it by generating a voltage that is small enough to avoid turning on semiconductor devices junctions.
For example, if an ohm function range uses a 1μA test current, then it can be used to measure resistors with a value of about
0.5V / 1μA = 500kΩ without turning on silicon PN junctions, which turn on at about 0.6V (you would need still smaller currents to measure the same resistance without turning on Schottky or Germanium diodes).
Such a tiny test current, however, when injected in a low resistance path will generate a really tiny voltage, which may be difficult to measure by the internal DMM circuitry.
E.g., the same 1μA current above will generate a scant 10μV voltage drop across a 10Ω resistor, voltage which is down in the noise in most situations and requires quite a sophisticated (and expensive) measurement circuitry.
Therefore, DMMs circuitry increase the test current when using ohm's ranges designed to measure low resistance values.
Since continuity testing is just meant to detect such low resistance values, (assuming the continuity function uses the same circuit used for resistance measurements) the test current is usually quite substantial (1mA is not uncommon).
Therefore, 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 into the circuit. This will turn on PN junctions, giving false readings for the resistance value.
For example, with a 1mA test current it is sufficient to have 500Ω of resistance across the probe terminals to generate a 500mV drop, which is enough to bring silicon diodes in conduction (and more so Schottky or germanium diodes). This will clamp the voltage to that value, making the DMM show you a 500mV/1mA = 500Ω value on the display.
If you want to check if this is really your case, try and measure with another DMM the test current put out by your original DMM on its continuity function. If it measures about 1mA you have a high chance that during the continuity measurement you turned on some semiconductor junction.
To show you what happens in a more practical way, here is a quick and dirty LTspice simulation of the following circuit, modeling the situation of a current source supplying current to a resistor in parallel with a diode:
The following are the results of a DC-sweep simulation with various test current values on the horizontal axis. The red curve is V(a), which is the voltage between the probes (what the DMM actually measures), and the green curve is the ratio between V(a) and the test current (that is, what the DMM actually displays).
As you can see, when the current is low enough, the DMM displays the correct value of 500Ω, i.e. that of the resistor R1. Once the current reaches about 0.7mA D1 begins to turn on, the voltage at A is clamped to about 0.6V and the displayed value decreases.
Now, since you said you measured 100kΩ with the resistance function, I was curious to reproduce it in the simulation, so I changed R1 to 100k. Here are the results:
which is pretty unreadable, but zooming in gives this picture:
I have added the yellow lines, which highlight that you would get a reading of 500Ω with a test current of about 1.2mA, which may be coherent with your situation.
I'd really like to hear you back telling us what test current value your DMM actually generates, as a definitive confirmation of my hypothesis.