Continuity Definition and 876 Threads

f(x) = sqrt(x+3), x > -3 3 x2-5, x < -3 first question would be limit x -> -3-, limit x -> -3+, limit x -> -3 the answer would be [-1, 1, undefined] But, i only got the 3rd answer correct, the first two are wrong?? but i don't get it why, the previous similar...

For the continuity equation (Q= Av, A is cross sectional area, v is velocity), is it independent of the radius of the pipe? If so, why?

Can anyone help me with this problem? Say f(x) & g(x) are cont. at x=5. Also that f(5)=g(5)=8. If h(x)=f(x) when x<=5 and h(x)=g(x) when x>=5: prove h(x) is cont at x=5.

Homework Statement Let f be a function with the property that every point of discontinuity is a removable discontinuity. This means that \lim_{y\to x} f(y) exists for all x, but f may be discontinuous at some (even infinitely many) numbers x. Define g(x) = \lim_{y\to x} f(y). Prove that g is...

Homework Statement Suppose f(x + y) = f(x) + f(y) and f is continuous at 0. Show that f is continuous at a. The attempt at a solution Since f is continuous at 0, for any e > 0 there is a d > 0 such that |f(x) - f(0)| < e for all x with |x - a| < d. Writing 0 as -a + a, |f(x) - f(0)| =...

How come sin(x^-1) is not continuous and xsin(x^-1) is?

1. The problem statement. Give a complete and accurate \delta - \epsilon proof of the thereom: If f is differentiable at a, then f is continuous at a. 2. The attempt at a solution Known: \forall\epsilon>0, \exists\delta>0, \forall x, |x-a|<\delta \implies \left|\frac{f(x) - f(a)}{x-a}...

So I am trying to derive the continuity equation: \frac{\partial}{\partial x^{\mu}}J^{\mu} = 0 From the Dirac equation: i\gamma^{\mu} \frac{\partial}{\partial x^{\mu}}\Psi - \mu\Psi = 0 And its Hermitian adjoint: i\frac{\partial}{\partial x^{\mu}}\overline{\Psi}\gamma^{\mu} -...

Hi, I was wondering if the Lebesgue-integral is continuos with respect to the L^\infty norm. More precisely, assume there is a space of functions \mathcal{P}=\{f\in L^1 :||f||_{L^1}=1\}\cap L^\infty endowed with the essential supremum norm ||\cdot||_{L^\infty}. If there is then a Cauchy...

Homework Statement Suppose that "f" satisfies "f(x+y)=f(x)+f(y)", and that "f" is continuous at 0. Prove that "f" is continuous at a for all a.Homework Equations In class we were given 3 main ways to solve continuity proofs. A function "f" is continuous at x=a if: a.) Limit of f(x) as x->a...

Hey folks, I have this one problem that seems unclear to me: Show that the function f is continuous at (0,0) for f(x,y) = ysin(1/x) if (x,y) do not equal (0,0)...and 0 if (x,y) = (0,0) I'm thinking though, as with single variable calc, f is continuous at...

Ok, i am hoping we can (owing to the large amounts of questions in the homework help on stuff like this) create a nice guide for Calculus in 3D including definitions, practise questions and general examples. Firstly, i know something i really do not like is Limits. My textbook gives the...

Homework Statement Let f be a function which is continuous on a closed interval [a,b] with f(c) > 0 for some c\in[a,b]. Show that there is a closed interval [r,s] with c\in[r,s]\subseteq[a,b] such that f(x) > 0 for all x\in[r,s]. Homework Equations Hint let epsilon = f(c)/2 and find...

Homework Statement Show, from the definition of continuity, that the power series function f(x)=sum(a_n*x^n) is continuous for its radius of convergence.Homework Equations Definition of continuityThe Attempt at a Solution Must show that for any |a| < R, given e>0 there exists d>0 such that...

Homework Statement Prove that the a continuous function with compact domain has a continuous inverse. Also prove that the result does not hold if the domain is not compact. Homework Equations The Attempt at a Solution I tried using the epsilon delta definition of continuity but...

Homework Statement Let f:R->R be a continuous function where limit as x goes to positive/negative infinity is negative infinity. Prove that f has a maximum value on R. Homework Equations None The Attempt at a Solution I tried to use the definition of infinite limits but I'm not...

Give an example of a function f which is discontinuous at 0 yet abs(f ) is continuous at 0 i have tried for an hour or so trying to think of one, even hints would be helpful

Let F(x) = Integral from a to x of f dt (a belongs in [a,b]) How do we show that F(x) is continuous? (f is Lebesgue integrable on [a,b] )

Homework Statement f(x) = {x^2 x \in Q -x^2 x \in R/Q At what points is f continuous? Homework Equations continuity: for every \epsilon > 0 there exists \delta > 0 d(f(x),f(p)) < \epsilon for all points x\inE for which d(x,p) < \delta The Attempt at a...

Let l \in R be the least upper bound of a nonempty set S of real numbers. Show that for every \epsilon < 0 there is an x \in S such that x > l - \epsilon I don't understand this question very well, I appreciate it if you could give me some hints. l is the l.u.b on S, therefore...

I'm trying to show a function has non-uniform continuity, and I can't seem to think of 2 sequences (xn) and (yn) where |(xn) - (yn)| approaches zero, where f(x) = x3. Can anyone think of two sequences?

1. a) Show that (f^-1 S)compliment = f^-1(S compliment) for any set S of reals. Then use part a) to show The function f is continuous iff f^-1(S) is closed for every closed set S. 2. inverse image = f^-1(S) = {x: f(x) \in S} f is continuous iff for every open set U \in the reals...

Any help on this problem would be appreciated a) Show that (f^-1 S)compliment is equal to f^-1(S compliment) for any set S of reals. Then use part a) to show The function f is continuous iff f^-1(S) is closed for every closed set S.

Hey guys, I'm in a little bit of a jam here: I managed to miss a really important lecture on continuity the other day, and there were a few examples that the professor provided to the class that I just got, but would love it if someone could explain them to me. First, f(x)=x3 is continuous...

Homework Statement The small piston of a hydraulic lift has a cross sectional area of (2.8 cm^2), and the large piston connected to it has an area of (17 cm^2). What force of F must be applied to the smaller piston to maintain a load of (28000 N)? Homework Equations A_1 * V_1 = A_2...

Homework Statement f(x) = 4 for x > or = 0, f(x) = 0 for x < 0, and g(x) = x^2 for all x. Thus dom(f) = dom(g) = R. Homework Equations a. Determine the following functions: f+g, fg, f o g, g o f. Be sure to specify their domain. b. Which of the functions f, g, f+g, fg, f o g...

Two problems, actually, but they are very similar. Here goes: Homework Statement Let f be a continuous real-valued function with domain (a, b). Show that if f(r) = 0 for each rational number r in (a, b,), then f(x) = 0 for all x in (a, b). Homework Equations The Attempt at a Solution...

Homework Statement Show that f(x) = x/(1+x^2) is continuous on R Homework Equations f is continuous at a if for any epsilon > 0, there exists a number delta > 0 such that if |x-a|<delta, then |f(x)-f(a)|<epsilon. The Attempt at a Solution |f(x) - f(a)| = |x/(1+x^2) - a/(1+a^2)|...

Homework Statement Prove: If f is defined on the reals and continuous at x=0, and if f(x1+x2)=f(x1)+f(x2) for all x1,x2 in the reals, then f is continuous at all x in the reals. Homework Equations Using defn of limits and continuity The Attempt at a Solution is this like proving that...

Let X be a norm space, and X=Y+Z so that Y\cap Z=\{0\}. Let P:X->Z be the projection y+z\mapsto z, when y\in Y and z\in Z. I see, that if P is continuous, then Y must be closed, because Y=P^{-1}(\{0\}). Is the converse true? If Y is closed, does it make the projection continuous? If...

Hi, I have a quick question: When we talk about smooth functions (say a vector field on Rn), why must we usually restrict the domain to an open set in Rn? Thanks!

Two questions need helps I got two questions below need helps: 1. Let f be a real continuous function defined on a closed subset E of R^1, then how can I prove the existence of some corressponding real continuous functions g on R^1, such that g(x)=f(x) for all x\inE ? 2. Let f and g two...

I am having trouble with the following proofs. If someone can help I would appreciate it. Problem Statement Given that f, g are continuous at z, prove that a- f+g is continuous at z b- For any complex \alpha, \alphaf is continuous at z There are other parts to this but if I think...

[SOLVED] ANALYSIS II: continuity of function in R^n Let A\subset\mathbb{R}^n and f:A\rightarrow\mathbb{R}. Show that, if the partial derivatives D_jf(x) exist and are bounded on B_r(a)\subset A, then f is continuous at a. We know...

Hi. How do I show that f is differentiable, but f' is discontinuous at 0? I guess I'm just looking for a general idea to show discontinuity. Thanks

Let f:\mathbb{R}^2\rightarrow\mathbb{R} be f(0,0)=0 and f(x,y)=\frac{x|y|}{\sqrt{x^2+y^2}} for (x,y)\neq(0,0). Is f continuous at (0,0)? I tried showing it WAS NOT continuous by finding sequences that converge to 0 but whose image did not converge to 0. I tried sequences of the form...

[SOLVED] Help with a pipe, water, continuity and Bernoullie Homework Statement At one point in a pipeline the water's speed is: 3.0 m/s, the gauge pressure is: 5.0*10^4. Find the gauge pressure at a second point in the line, 11m lower than the first, if the pipe diameter at the second...

[SOLVED] Uniform Continuity Homework Statement Let A \subset \mathbb{R}^n and let f: A \mapsto \mathbb{R}^m be uniformly continuous. Show that there exists a unique continuous function g: \bar{A} \mapsto \mathbb{R}^m such that g(x)=f(x) \ \forall \ x \in A . Homework Equations...

I need to prove that for every continuous function f:X->X of a metric and compact space X, which satisfy for each two different x and y in X p(f(x),f(y))<p(x,y) where p is the metric on X, there's a fixed point, i.e there exist x0 s.t f(x0)=x0. obviously i thought assuming there isn't such a...

[SOLVED] hyperplanes and continuity Homework Statement Let X be a real normed linear space, f a linear functional on X and c a real constant. The set f^{-1}(c) is called the hyperplane of equation [f=c] and supposedly, the hyperplane of equation [f=c] is closed if and only if f is continuous...

Homework Statement The problem is from Stewart, Appendix G, A58, no.45. Suppose that F, G, and Q are polynomials, and: F(x)/Q(x) = G(x)/Q(x) for all x except when Q(x) = 0. Prove that F(x) = G(x) for all x. [Hint: Use Continuity] The Attempt at a Solution I thought the statement was...

Homework Statement Find the constants a and b such that the function is continuous on the entire real line. Homework Equations f(x)={2, x< or = -1 {ax + b, -1<x<3 {-2, x> or = 3 The Attempt at a Solution I don't know where to start. If anyone is willing to help...

Homework Statement Find the constants a and b such that the function is continuous on the entire line. Homework Equations g(x)={4 sinx/x, x<0 {a-2x, x> or = 0 The Attempt at a Solution Possible discontinuity at x=0 f(0^+)=a-2x=a-2(0)=a f(0^-)=4sinx/x=4sin(0)/(0) i am...

[SOLVED] A formulation of continuity for bilinear forms Homework Statement My HW assignment read "Let H be a real Hilbert space and a: H x H-->R be a coninuous coersive bilinear form (i.e. (i) a is linear in both arguments (ii) There exists M>0 such that |a(x,y)|<M||x|| ||y|| (iii) there...

[SOLVED] River channel problem using Bernoulli and Continuity Homework Statement A river (100 m wide) flows through its rectangular channel at a depth of 2.560 m at a velocity of 2.050 m/s. What is the velocity of the discharge if the channel is narrowed to 90 m? Homework Equations...

So I'm trying to grasp the epsilon,delta definition of limits.(Well not really,I'm actually just trying to be able to get the majority of the related questions right.) For example: when taking limits of rational functions: A result of 0/0 is indeterminate form(suggesting a hole in...

Homework Statement The aorta carries blood away from the heart at a speed of about 40 cm/s and has a radius of approx. 1.1cm. The aorta branches eventually into a large number of tiny capillaries that distribute the blood to the various body organs. In a capillary, the blood speed is approx...

1.suppose that f:X->Y is continuous. if x is a limit point of the subset A of X, is it necessarily true that f(x) is a limit point of f(A)? 2. suppose that f:R->R is continuous from the right, show that f is continuous when considered as a function from R_l to R, where R_l is R in the lower...

Homework Statement I'm reading this at the moment: "Let f:R^n-->R^n be of class C^1 (that is, assume Df exists and is continuous)" What does it mean?? If it means that for all x in R^n, the linear map Df(x):R^n-->R^n is continuous, then it's a triviality since all linear maps from R^n to R^m...

Homework Statement Use the Schrodinger Equation to show that \frac{\partial}{\partial t}(\Psi^{*} \Psi) = - \underline{\nabla}. \underline{j} Homework Equations \underline{j} = \frac{-i}{2m} \left[\Psi^{*}(\nabla \Psi) - (\nabla \Psi^{*})\Psi]\right \frac{\partial}{\partial...