Continuity Definition and 876 Threads

  1. S

    Continuity and Intermediate Value Theorem

    Hello everyone, I have come across two questions that I have solved, but unfortunately am quite sure I've done them incorrectly. They are related to continuity and the intermediate value theorem. Find the constant c that makes g continuous (-infinity,infinity). g(x){ x^2-c^2 if x<4...
  2. A

    Prove Determinant of Rotation Matrix is 1 w/Continuity Argument

    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?
  3. D

    Finding Roots and Continuity of Functions

    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...
  4. A

    [calculus] Continuity of partial derivatives

    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...
  5. K

    Lipschitz Continuity and Uniqueness

    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...
  6. M

    Can you help me understand limits and continuity in Calculus?

    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...
  7. S

    How Can You Prove Uniform Continuity on a Closed Interval?

    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...
  8. S

    Is a Continuous and Periodic Function Bounded and Uniformly Continuous on R?

    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...
  9. K

    Solving the Continuity Equation

    i do not understand how the continuity equation works?
  10. D

    Having trouble with limits and continuity? Let's clear things up!

    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...
  11. M

    Can You Find a Continuous Function That Takes Each Value Exactly Three Times?

    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...
  12. M

    Are My Definitions of Continuity and Uniform Continuity Correct?

    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...
  13. S

    Can Two Discontinuous Functions Become Continuous When Added Together?

    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...
  14. M

    Calc Proof(s): Uniform Continuity

    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...
  15. D

    Continuity of a Function of Two Variables

    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) =...
  16. T

    Proving Continuity with Epsilon-Delta: How to Approach a Challenging Function?

    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...
  17. Z

    Uniform Continuity (and Irrationals)

    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...
  18. U

    Find Value of K for Continuity of f(x) at x=1

    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...
  19. E

    Help on Continuity - Finding Nonremovable & Removable Discontinuities

    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...
  20. H

    Solve for K for Continuity of f(2) at @=0

    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...
  21. Clausius2

    What is the solution for the Continuity Equation at r=0?

    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...
  22. D

    How Do I Prove Continuity for f(x)=x^2 at x=3 and for f(x,y)=1/(xy)?

    How do I show that f(x)=x^2 is continuous at any given point, say x=3. Thank you.
  23. himanshu121

    Continuity And Differentiability

    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...
  24. phoenixthoth

    The notion of continuity applied to sets

    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...
  25. T

    Time and Indeterminacy vs. Continuity

    '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...
  26. P

    Understanding Continuity and Discontinuity in Calculus

    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...
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