Continuity

The question doesn’t give any context, so, since this is about continuity my guess is that it involves a mathematical definition of continuity. Let’s say there’s a function represented by f(x). For example, if y=3x²+4, then f(x)=3x²+4. This function is continuous because if we change x by a small amount, y changes by a small amount, no matter what x actually is. So the value of x is arbitrary. We can represent this arbitrary value as “a”. So, when x=a, y=3a²+4 or f(a)=3a²+4. And we can also say that if x is near to a, or, the limit of f(x) as x approaches a (normally written as Lim as x→a) is 3a²+4. If a=2, then this limit would be 16.

Now we come to the bit you’re having a problem with. If we don’t know a how can we work this out? The answer is, we don’t have to know a because a is arbitrary (a = arbitrary). It’s part of the domain (all values that x can take) so any value will do, because we are testing to see if the function itself is continuous over all the values x can take. What we’re saying is: a function is DEFINED as being CONTINUOUS, IF, as x approaches an ARBITRARY value a, f(x) approaches a certain value L. In the example L would be 16 if a=2; but I could have picked a=1, so L=7; if a=100, L=30004. We don’t need to put an actual value for a or L, because we are talking about the function as a whole—continuity is a property of a function, not of any particular part of it.

Now, consider the opposite of continuity: discontinuity. For example, f(x)=1/(1-x). It’s discontinuous because it’s not true that for ANY arbitrary value a there is a certain value L for which we can say Lim as x→a f(x)=L. Why? Because if a=1, we can’t define f(x). Other values of a are OK, but this one value a=1 doesn’t work, so the function isn’t continuous. It also means that, since we can’t calculate f(1), a small change in x doesn’t equate to a small change in f(1) (close to L). In other words we can’t define L.

I hope this resolves your confusion.

by Top Rated User (775k points)