A false continuum argument makes the claim that because a continuum exsist between two extremes, there is no real distinction between the extremes. For example, there are many shades of grey, therefore everything from black to white is nothing but a shade of grey, and the concepts of black and white are irrelevant. This is a false continuum because it is still usefull to distinguish between black and white. This is used to associate, or to blur the distinctions between, things that should be seen as distinct.

False continuums are closely related to the Sorites Paradox - understanding one helps you understand the other.

If you have one grain of sand, it's not a heap. If one grain of sand doesn't make a heap, then two do not. If two grains of sand don't make a heap, then three do not. You can keep on going, until you eventually get to the point where you have 50,000 grains of sand and you still don't have a heap.

Now, you can bite the bullet and deny that heaps exist, or you can argue that there is some intrinsic vagueness in the problem. There are various other responses described in the Stanford Encyclopedia of Philosophy article on the Sorites Paradox.

Similar problems exist when it comes to understanding, say, probabilities or statistics. If someone said to you that a drug works for 99% of patients, you might decide that it's probably worth taking the risk. But if you do that, then surely you should also take the risk if it works for 98%, 97%, 96% etc. Where you draw the line to say it is 'effective' is pretty arbitrary. But that doesn't mean that there isn't a meaningful difference between a drug that works for 99% of people and 43%, just as a heap existing being meaningful, even if the truth conditions for something being a heap are vague. In the real-world example, there might be other factors: there is the risk comparison, there are personal factors, economic factors, comparison with other medical interventions and so on.

because i can balance one biscuit on my hand and i have space for 50 more i can balance 50 biscuits on my hand

would be an example

To add to the above answers, the Sorities paradox is relevant to effecting change in society, in business, the spreading of disease and even down to trying to make your band famous.

It's all about critical mass. The Sorities paradox is not actually a paradox and depends on the viewer. If you watch fromt he start you may not see (to contiue the above example) "a heap", but if you enter part way through you will instantly declare the rice is a heap.

The continuum therefore only exists to those inside and is a useful tool in effecting change in that in making changes, if you do so in small doses, those on the inside will not notice the change as they do not notice the heap.

For example, displace a small village and it is a small regretable side-effect of a larger benefit; do this many many many times and they are all small regretable side-effect until you have reached critical mass and reversal cannot occur without now worse upheaval.

Hope this helps