Continuity Definition and 876 Threads

Homework Statement Refer to attached file. The attempt at a solution (a) g'(0) = \lim_{x\rightarrow 0} {\frac{g(x)-g(0)}{x-0}} g'(0) = \lim_{x\rightarrow 0} {\frac{x^\alpha cos(1/x^2)-0}{x}} g'(0) = \lim_{x\rightarrow 0} {x^{(\alpha-1)} cos(1/x^2)-0} g'(0) = 0 So...

Where is the function $${f (x, y) = x^2 + x y + y^2}$$ continuous? How do I go about solving such problems?

Homework Statement To study the continuity of a function with two variables (x,y). Homework Equations f(x,y)=\frac{x^3}{x^2+y^2} if (x,y)\neq(0,0) f(x,y)=0 if (x,y)=(0,0) The Attempt at a Solution I've tried going by the composition of functions but I can't seem to get anywhere...

I don't understand uniform continuity :( I don't understand what uniform continuity means precisely. I mean by definition it seems that in uniform continuity once they give me an epsilon, I could always find a good delta that it works for any point in the interval, but I don't understand the...

Homework Statement (X,d1),(Y,d2) and (Z,d3) are metric spaces, Y is compact, g(y) is a continuous function that maps Y->Z with a continuous inverse If f(x) is a function that maps X->Y, and h(x) maps X->Z such that h(x)=g(f(x)) Show that if h is uniformly continuous, f is uniformly...

Homework Statement 2. Show that the function is continuous on the given interval. (a)f(x)= (2x+3)/(x-2) range:(2, infinity) (b)f(x) = 1- sqrt(1-x^2) range:[-1,1] 3. Prove that the following limits do not exist. (a) lim x tends to 0 ( absolute|x|/x) (b) lim x tends to 3 (2x/(x-3))...

Homework Statement Suppose that a and b are real numbers. Find all values of a and b (if any) such that the functions f and g, given by a) f(x)={ax+b if x<0 and sin(x) if x≥0} b) g(x)={ax+b if x<0 and e2x if x≥0} are (i) continuous at 0 and (ii) differentiable at 0...

Homework Statement Homework Equations The Attempt at a Solution Basically the first part of the question asks to prove the binomial theorem through induction which I've done. I'm basically lost as to how to even attempt these questions, I'm not asking for answers as I need to know this...

Homework Statement This is a problem from my Analysis exam review sheet. Let L(x) = \sqrt{x}. Prove L is continuous on E = (0,\infty) The Attempt at a Solution The way we've been doing these proofs all semester is to let \epsilon > 0 be given, then assume \left| x -x_{0} \right| <...

I'm working on a problem as part of exam revision, but I've run into a bit of trouble so far. The problem is; Give an (ε,δ) proof that f(x) = 1/\sqrt{10 - x^2} is continuous at x = -1 The attempt at a solution So far what I've gotten is f(x) - f(-1) = 1/(\sqrt{10 - x^2}) - 1/3 = (3 -...

I can't seem to wrap my head around this concept, I'm hoping you can help me out. Suppose you have a continuous function defined on some compact subset of the plane, say {0 <= x <= 1, 0 <= y <= 1}. I guess the function could be either real or complex valued, but let's just say it's real so we...

I'm seeking a neater proof for the following: Let E \subseteq R. Let every continuous real-valued function on E be bounded. Show that E is compact. I tried to argue based on Heine-Borel theorem as follows: E cannot be unbounded because if it is the case, define f(x)=x on E and f(x) is...

Homework Statement Let f:[0,1] →ℝ be a continuous function that does not take on any of its values twice and with f(0) < f(1), show that f is strictly increasing on [0,1]. Homework Equations The Attempt at a Solution Assume that f is not strictly increasing on [0,1]. Therefore...

Homework Statement Let I:=[0,1], let f: I→ℝ defined by f(x):= x when x is rational and 1-x when x is irrational. Show that f is injective on I and that f(fx) =x for all x in I. Show that f is continuous only at the point x =1/2 **I think i addressed all of these questions but I am unsure...

Homework Statement If the function f+g:ℝ→ℝ is continuous, then the functions f:ℝ→ℝ and g:ℝ→ℝ also are continuous. Homework Equations The Attempt at a Solution Ok, just learning my proofs here, so I'm not sure if my solution is cheating or not rigorous enough. take f(x)= {-1 if...

Homework Statement Prove that if F is uniformaly continuous on a bounded subset of ℝ, then F is bounded on A. Homework Equations The Attempt at a Solution F is uniformaly continuous on a bounded subset on A in ℝ. Therefore each ε>0, there exists δ(ε)>0 st. if x, u is in A where...

Homework Statement Suppose a function f is continuous and has continuous derivatives of all orders for all x. it satisfies xf ''(x) + f '(x) + xf(x) = 0. Given f(0) = 1 find the value of f '(0) and f '' (0). Homework Equations The Attempt at a Solution when x=0, 0f''(0) + f ' (0) + 0f(0)...

If the function f:D→ℝ is uniformly continuous and a is any number, show that the function a*f:D→ℝ also is uniformly continuous. Ok, so I am just learning my proofs so be patient with me, I'm very new at it. take a>0, ε>0 and x,y in D. We know |x-y|<δ whenever |f(x)-f(y)|<ε. If we take...

Homework Statement I want to make sure that a solution to a differrential equation given by bessel functions of the first kind and second kind meet at a border(r=a), and it to be differenitable. So i shall determine the constants c_1 and c_2 I use notation from Schaums outlines Homework...

Homework Statement Determine where the function f(x + iy) = 2sin(x) + iy^2 + 4(ix - y) is differentiable and where it is analytic.The Attempt at a Solution f(x + iy) = 2sin(x) -4y + i(y^2 +4x) Through C-R equations: du/dx = 2 cos x dv/dy = 2y du/dy = -4 dv/dx = 4 So the C-R equations hold...

Is the function $f(x) = \left\{\begin{array}{rcl}\sqrt{x}\cos\left(\frac{1}{x}\right)&\text{if}&x\neq 0\\0 &\text{if}&x=0\end{array}\right.$ continuous at 0? My answer is no, because the left hand limit does not exist. Am I right?

Homework Statement Let f be continuous on the interval [0,1] to ℝ and such that f(0) = f(1). Prove that there exists a point c in [0,1/2] such that f(c) = f(c+1/2). Conclude there are, at any time, antipodal points on the Earth's equator that have the same temperature. Homework Equations...

I'm a new poster; I hope don't violate rules or policy. Am I wrong, or is the rule essentially that a particle isn't finished interacting with the last particle on it's world path unless and until it interacts with a third? This isn't what I might have guessed about physical continuity, but it...

Homework Statement Let E be a dense linear subspace of a normed vector space X, and let Y be a Banach space. Suppose T0 \in £(E, Y) is a bounded linear operator from E to Y. Show that T0 can be extended to T\in £(E, Y) (by continuity) without increasing its norm. The Attempt at a Solution...

Homework Statement Show that there exist nowhere continuous functions f and g whose sum f+g is continuous on R. Show that the same is true for their product. Homework Equations None The Attempt at a Solution Let f(x) = 1-D(x), where D(x) is the Dirichlet function Let g(x) = D(x)...

What does it mean if the continuity equation equals a complex number (rather than zero)? I ask this in the context of the probability current.

Homework Statement x*sin(sqrt(x^2+y^2))/sqrt(x^2+y^2) find the domain of continuity Homework Equations none The Attempt at a Solution I found the domain, which is x^2+y^2 > 0 and since x^2 >= 0 and y^2 >= 0 therefore the domain is (-inf,0) (0,inf) but the professor then asked...

consider this function f(x)=[x[\frac{1}{x}]] ([x] represent greatest integer less than or equal to x or in short GIF ) internal brackets over 1/x and external brackets are around full body of function. discuss on these points(means either are these correct incorrect) Statement 1: this function...

Is this claim true? Assume that X,Y are topological spaces, and that all closed subsets of X are compact. Then all continuous bijections f:X\to Y are homeomorphisms. It looks true on my notebook, but I don't have a reference, and I don't trust my skills. Just checking.

Continuity equation is dj+\partial_t\rho_t=0 where j and \rho are a time-dependent 2-form and a time-dependent 3-form on the 3-dimensional space M respectively. (see e.g. A gentle introduction to the foundations of classical electrodynamics (2.5)) If we use differential forms on the...

Homework Statement Prove that if f is continuous at a, then for any ε>0 there is a σ>0,? such that if abs(x-a)< σ and abs(y-a)< σ then abs[f(x) - f(y)]< ε Homework Equations Definition of continuity and triangle inequality abs(f(x)-f(y))= abs(f(x)-f(a) + f(a)-f(y))≤ abs(f(x)-f(a))+...

Homework Statement A ball falls from rest at a height H above a lake. Let y = 0 at the surface of the lake. As the ball falls, it experiences a gravitational force -mg. When it enters the water, it experiences a buoyant force B so the net force in the water is B - mg. a) Write an...

How can I prove the below is continuous? $$ \int_{-\pi}^{\pi}te^{xt}\cos(yt)g(t)dt \quad\text{and}\quad -\int_{-\pi}^{\pi}te^{xt}\sin(yt)g(t)dt $$ define the Fourier transform of g as $$ G(z) = \int_{-\pi}^{\pi}e^{zt}g(t)dt $$ We know t, e^{xt}, sine, and cosine are continuous which means...

Show that the following are continuous at x=1 using the epsilon-delta definition: $x^{2} - x + 1$ $\sqrt (x)$ I know the definitions but I don't really know quite what to do with them. After the simple rearranging I'm just at a bit of a dead end; any pointers? Gracias, GreenGoblin

Homework Statement The random variable X has the binomial distribution B(20,0.4), and the independent random variable Y has the binomial distribution B(30,0.6). State the approximate distribution of Y-X, and hence find an approximate value for P(Y-X>13) Homework Equations The...

By continuity I mean an unbroken fractal. With certain variants, one ends up with sharp gaps in the fractal. mag=({x^2+y^2+z^2})^{n/2} yzmag=\sqrt{y^2+z^2} \theta= n *atan2 \;\;(x + i\;\;yzmag ) \phi = n* atan2\;\; (y + iz) new_x= \cos{(theta)}\;*\;mag new_y=...

Homework Statement use delta, epsilon to prove that e^x is continuous at c = 0 Homework Equations (a) for y>0, lim_n-> inf, y^(1/n) = 1 (b) for x < y, exp(x) < exp(y) The Attempt at a Solution im not sure how to approach this problem. i have, |exp(x) - exp(0)|= |exp(x) - 1|...

Can anyone give me an example of a continuous function that is NOT differentiable(other than the square root function)? I have to prove that not all continuous functions are differentiable. Thanks!

Homework Statement For each of the following functions, find a value of a, (if such a value exists), which makes the function continuous. a) f(x) = {ax^2...x > 3 ...{x - 7...x ≤ 3 b) f(x) = {sin(ax)...x < (pi) ...{1...x ≥ (pi) c) f(x) = {x^2 + a^2...x > 1 ...{9 - x....x ≤ 1...

Sup guys, I was just going over my Baby Rudin and I came across a problem that I don't really know how to get started on. Suppose f is a real function defined on R that satisfies, for all x Limit_{n\ \rightarrow \ 0} (f(x+n)-f(x-n)) = 0, does this imply f is continuous? My first thoughts...

Homework Statement Hey guys, I have two separate questions. 1.) I am asked for a unit vector pointing from P = (1,2) to Q = (4,6) In physics, every vector I've ever worked with started at the origin, so these feel weird. I initially thought that it would simply be 3i + 4j, the...

Homework Statement Homework Equations Ability to graph functions. Essential Discontinuity: Jump discontinuity or Infinite discontinuity The Attempt at a Solution First Question: After plugging in 2 for every equation and getting a result that was greater than 0, I...

Homework Statement Let \begin{equation*} f(x,y) = \begin{cases} \dfrac{x^3 - y^3}{x^2 + y^2}, \hspace{1.1em} (x, y) \neq (0,0) \\ 0, \hspace{4em} (x,y) = (0,0) \end{cases} \end{equation*} Is f continuous at the point (0,0)? Are f_x og f_y continuous at the point (0,0)? Homework Equations...

Homework Statement A large vertical cylindrical rainwater collection tank of cross sectional area A is filled to a depth h. The top of the tank is open and in the centre of the bottom of the tank is a small hole of cross sectional area B (B<<A). Derive expressions for (i) the flow speed...

my book says, the function y = x*sin(1/x) is not continuous at x = 0, however by defining a new function by F(x) = x*sin(1/x) , x ≠ 0 0 , x = 0 then F is continuous at x = 0. This does not make sense to me because the limit as x → 0 is equal to 1, not zero...

Why does the book say if f(x) is continuous at a then lim ( f(a + Δx) - f(a) ) / Δx, that Δx will go to zero. What does that mean geometrically? Δx→0 More importantly, why would Δx not approach zero if f(x) is not continuous at a? Im guessing it has something to do with the slopes of...

Since I am new to PF (hi!), before I go any further, I would like to a) briefly note that this is an independent study question, and that its scope goes beyond that of a textbook question - i.e., I believe that this thread belongs here - and b) also note that I am new to analysis and early...

A function f is defined as follows: f(x) = 2cos(x) if x≤c, = ax^2 + b if x > c . Where a,b, and c are constants. If b and c are given. find all values of a for which f is continuous at the point x = c Solution: a = (2cos(c) - b)/c^2 if c ≠ 0 ; if c = 0 there is...

Can anyone suggest me a self study book for limits and continuitsy whic starts from zero level to a very advanced level thanks in advance rithu

thanks!