Let c be in S subset R and f: S->R be a function. Assume that for every sequence
ID: 3080849 • Letter: L
Question
Let c be in S subset R and f: S->R be a function. Assume that for every sequence {Xn} subset S with the property that lim Xn=c the sequence {f(Xn)} is convergent. Show that f is continuous at c. I need help with a formal proof. Perhaps using the definition of continuity and covergence.Explanation / Answer
I assume that you are mostly interested in functions where the input and output are both real numbers, and the domain is an interval. (I'm only going to talk about those functions in my answer.) You're probably familiar with the idea that continuous functions are supposed to have graphs without jumps--that is, graphs that you can draw without lifting your pen from the paper. Uniform continuity means the same thing as continuity, plus one additional condition: There must not be a situation where the value of the function is changing ever more and more rapidly as you move towards the edge of the domain. To understand why a function could fail to be uniformly continuous, let's look at a specific function. The function f: (0, 8) -> R defined by f(x) = sin (1/x) is a good one. This function is continuous on (0, 8), but not uniformly continuous. You should graph it on a graphing calculator to get a sense for what the graph looks like. Or use Wolfram Alpha: http://www.wolframalpha.com/input/?i=sin… (Remember that I'm only considering x > 0, so if you use my link, ignore the left half of the graph.) Notice how the graph has no jumps in it (so the function is continuous). However, as you approach x = 0 from the right, something unusual happens--the graph starts oscillating between y = -1 and y = 1 faster and faster and faster (in fact, before you get to 0, the graph oscillates infinitely many times). This sort of odd behavior towards the edge of the domain is exactly what prevents a continuous function from being uniformly continuous. Note also that lim x->0+ [sin (1/x)] does not exist. This is not an accident. *Any* function which is continuous but not uniformly continuous *must* have one of the limits of the function as you approach the edge of an interval fail to exist. (Note that having a limit be 8 or -8 also counts as the limit failing to exist.) One more example: Consider the function f: (-8, 8) -> R defined by f(x) = x^2. (I assume that you are quite familar with the graph of this function.) Again, note that as we move towards the edge of the domain (in this case, towards x = -8 or towards x = 8), the function grows faster and faster and faster. So this function is also continuous but not uniformly continuous. Again, note that lim x->-8 [x^2] and lim x->8 [x^2] both fail to exist (they're 8). --- Some more comments to help illustrate the difference: * The only condition uniform continuity really adds to continuity is that the behavior towards the edge of the domain does not involve ever-more-rapid changes in the value of the function. (Any continuous function is automatically uniformly continuous on any closed interval [a, b] which is a subset of its domain; it really is only at the very edge of an open interval that you can mess up uniform continuity in any way other than a jump in the graph.) * As a special case of the previous comment: Any continuous function f: [a, b] -> R is automatically uniformly continuous. (Only functions where at least one side of the interval is open, or where one side of the interval is 8 or -8, can be continuous but not uniformly continuous.) * A function with a bounded derivative (so that the function is continuous, and its value never rises or falls more quickly than at some fixed rate) is automatically uniformly continuous. Note that all of the comments I've made are just to provide you with an idea of what "uniform continuity" means; they're not necessarily all exactly equivalent to it. As usual in math, if you're trying to decide whether something is uniformly continuous or not, you should see whether it satisfies the definition. (But perhaps some of my ideas might allow you to make a more educated guess as to whether a function is uniformly continuous.) --- The idea of uniform continuity is important because there are many theorems about uniformly continuous functions; so if you can prove that a function is uniformly continuous, then you already know a great deal about it (much more than you would know if you just knew it to be continuous). These theorems might allow you to compute integrals or derivatives more easily in certain situations, for example, either of which has an immense number of real-world applications.
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