Continuity Definition and 875 Threads

What's a continuity argument? For example, a question asks to prove that the determinant of a rotation matrix is always 1 using a continuity argument?

I recently finished a homework assignment with the exceptions of the following: 1.) f(x) =x^3 - x^2 + x, show there is a number c such that f(c)=10. f(x) can be equated to 10, but I'm not quite sure how to solve the equation from that point. 2.) Prove that the equation has at least...

Hello, If I am given a function of several variables and a parameter. Such as: f(x,y,z)=\frac{x y z^2}{(x^2+y^2+z^2)^k} This function is defined to be 0 where it is incontinuous (in (0,0,0)). How can I conclude for which values of k the function has three continuous partial derivatives? I...

Dear all, If a differential equation is Lipschitz continuous, then the solution is unique. But what about the implication in the other direction? I know that uniqueness does not imply Lipschitz continuity. But is there a counterexample? A differential equation that is not L-continuous, still...

Hello everyone: I am a new member of this forum and this is my first post. I was referred by an Astronomy.com member and so I decided to check it out. First I would like to introduce myself. My name is John and I am seventeen years of age and a senior in High School. My Calculus teacher...

hello all i have been working on this problem, see i can see how it could be true but i don't know how to prove it,would anybody have any ideas on how to prove this ? let [a,b] be a closed interval in R and f:[a,b]->R be a continuous function. prove that f is uniformly continuous...

hello all, i have been working on problems with continuity and i have come across some question in which i understand generally what i have to do but i just don't know where to start and how to put it together a function f:R->R is said to be periodic if there exists a number k>0 such...

i do not understand how the continuity equation works?

I'm having trouble with limits that involve 0+ and 0-. Can someone show me how the answers to the following limits are obtained? f(x) = \frac{1}{1+e^{\frac{1}{x}}} \lim_{x\rightarrow0^{+}} = 0 \lim_{x\rightarrow0^{-}} = 1 Now, my second query involves continuity. I understand that...

I am having great, great difficulties in solving this problem, its asking me to find a function that is continuous everywhere which takes each of its values exactly 3 times(like give an example of a function, no proving). This part, i have a little imagination of my own to start, but the second...

I'm seeking a bit of affirmation or correction here before i try to solidify this to memory... I know continuity to mean: Let f:D -> R (D being an interval we know to be the domain, D) Let x_0 be a member of the domain, D. This implies that the function f is continuous at the point...

just a basic question, so if I'm asked to find 2 functions that are discontinus, but when added together, becomes continuous, how do I approach that? can I say like, let F(x) = 1 for x =< 0, and f(x) = 0 for x > 1. G(x) = 1 for x =<0 and g(x) = 0 for x = 1. can I just somehow "add" f...

There are two proofs which I have attempted to work on that have beein somewhat trifling. The first of which is : prove that the function f: [0,infin) -> R defined by f(x) = 1/x is not uniformly continuous on (0,infin). im thinking that the way in which i should probably attempt to solve...

Question: Is the function f(x,y) = (x^2 - y^2)/(x-y) continuous at (1,1) if we set f(1,1) = 0? Why or why not? So far, I've just plugged 1 in for x and y and found the limit to equal 0. I guess that means that the limit is not continuous at (1,1)? And what do they mean by set f(1,1) =...

Hi I am trying to prove the continuity of a function. I do understand the definition and I can do it for "smaller" functions. However, for this "larger" function I am having troubling bounding it and thus can't find a prove. Any suggestions would be greatly appreciated! Show, using the...

I am a little shaky with the concept of proving uniform continuity vs regular continuity. Is the difference when proving through epsilon-delta definition just that your delta can not depend on "a" (thus be defined in terms of "a") (when |x-a|<delta) for uniform continuity? Also to the more...

let f be a function defined by f(x)=4-x^2 when x=<1 and k+x when x>1 What value of k will f be continuous at x=1? I know the answer is k=2, however, I don't know how to show to correct work. I got 2 when I sketched a graph of 4-x^2 and plugged in some numbers but I don't know how to show it...

Help on continuity! 1) Ok, i know how to find x, but how do you know if something is nonremovable or removable discontinuity? like for this: f(x)=|x+2|/(x+2) i knoe its x=-2, but is it nonremovable or removable? 2) How do u do the continuity stuff with there: f(x)=csc2x...

Hi, I have just one more problem :) here it is: (because I don't know how to do theta, "@" will equal theta) Find a value of K so that f(2) is continuous at @=0 ( != means "not equal") f(@) = ( (2sin@)/@ , @ !=0 ) ( 5k , @=0 ) f(@) is a piecewise function I don't know...

Hi guys. I am solving the axisymmetric free jet of an incompressible fluid. But I have troubles at r=0. Continuty equation can be written in cylindrical coordinates as: 1/r*d(rv)/dr + du/dz=0 v=radial velocity (v=0 at r=0) u=axial velocity. hz=delta(z) hr=delta(r) What happens at...

How do I show that f(x)=x^2 is continuous at any given point, say x=3. Thank you.

Consider f(x)=x^3-x^2+x+1 g(x)=\left\{\begin{array}{cc}{max\{f(t),0\leq t \leq x\}}\;\ 0\leq x \leq 1 \\ 3-x\;\ 1< x \leq 2\end{array}\right Discuss the continuity and differentiability of g(x) in the interval (0,2) I know how to do it As f(x) is increasing function therefore max...

i'm trying to define what it would mean for a set to be continuous. what i'd like to say is that S is continuous if it is homeomorphic to [0,1], (0,1], or (0,1). (perhaps that's redundant already?) but I'm not sure if that captures all the sets i'd like to think of as continuous. my...

'Time and Indeterminacy vs. Continuity' Sorry; I couldn't resist that. (Peter Lynds theory) What is time? Dictionary.com/time: 'Time is a nonspatial continuum in which events occur in apparently irreversible succession from the past through the present to the future. An interval separating...

Is it just me or is it the text didn't explain well enough about continuity? I totally don't understand it. Not only that, when it covers about derivatives it first introduced another defination of slope. Then it says that the derivative of x^2 is 2x. How does slop relate to derivative? Can...