# Is the function f(x)=sin(1/x) continuous, uniformly continuous, or discontinuous on (0,2 pi)?

De ne f n: [0;2] !Rby f n(x) ... f(x) = sin x2. (ii) f(x) = inffjx n2j: ... converges pointwise to a continuous function f, but f is not uniformly continuous. [Hint: ... - Read more

Hence the Fourier series of F converges uniformly to the ... 1(x) = x−(1/2) is sectionally continuous. ... Compute the Fourier series of the function f(x) = ... - Read more

## Is the function f(x)=sin(1/x) continuous, uniformly continuous, or discontinuous on (0,2 pi)? resources

### Absolute continuity - Wikipedia, the free encyclopedia

... (x) over the entire real line, and sin(1/x) ... Every absolutely continuous function is uniformly ... A function f: I → X is absolutely continuous on I if for ...

### Uniform Continuity - University of Kansas

0 = 2, we require (4 + ) < . And if x ... (x) f(y)j= 1 x 1 y = jx yj ... The function f(x) = 1=xis uniformly continuous on the closed interval [1=2;1].

### Continuous functions - An approach to calculus

... "The function f(x) is continuous at x = c."? ... That function is discontinuous at x = c. ... "The function y = 1 x: is continuous for all values of x in its domain."

### Section 5.3, Problem 2 - UCSD Mathematics | Home

1(x) = 1 √ π cos(x)+cos(2x). ... This is because the function f(x) = x3 is only discontinuous at the ... This is because the function f(x) = ‘x − x2 is ...

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4−x2 − 2y 2x +2y ≤ 4 [0,2] f(x,y) = 1/(x2 +y ... A function f(x,y) is called continuous at ... Sketch the graph and contour map of the function f(x,y) = sin(x2 ...

### Math 3001 Due Wed Nov 9 Assignment #7

on D for all n and x, but f(x) is discontinuous at x = 0. 2. ... verges uniformly to a function f that is continuous on [0,1]. ... (1+x) ln (1−x) = 2 X∞ n=0 x2n ...

### Fourier Transforms - Massachusetts Institute of Technology

R Deﬁnition. f ∈ L1(R) deﬁne the fourier transform fˆby fˆ(y) = f (x)e−ixydx, y ∈ R R fˆis bounded and uniformly continuous ...

### Taylor and Maclaurin - Scarlet

e x and its Maclaurin polynomial We know that if f(x) = e x then f (n) (x) = e x and f (n) (0) = 1 Thus T n e x = 1 + x/1! + x 2 /2! + x 3 /3! + ... + x n /n! sin(x ...

### Differentiable and Continuous Worksheet | Problems & Solutions

Every continuous function need not be differentiable and every differentiable function is continuous. ... = sin 2 x if 0 < x ≤ π 4 ... + f (x), the function f (x ...

### Final Exam SAMPLE (with answers and hints) - www.math.uci.edu

... A continuous but not uniformly continuous function f : ... (and discontinuous at every ... f(x) = x2 sin 1 x2 is uniformly continuous on (0;5).

### Solution Set Math 205a - Fall 2011 Problem Set 1

In either case we rst prove it under the assumption that f is continuous ... = x2 sin(1=x) and g(x) = x2 sin ... Problem 9 Show that an absolutely continuous function ...

### Continuity - Courant Institute of Mathematical Sciences

f(x) = 1 1−x2 is discontinuous at x = ... The function g is continuous at a = 0;2;5: In fact, (i) ... 6. f(x) = sinx+ √ x− 1 x ...

### Tier 1 Analysis Exam: August 2011 1. 2 X 2 X f 2

Let f: X ! X be continuous. Fix a point x0 2 X, ... fn+1(x) = Z x a fn(t)dt; ... (x) converges uniformly on [a;b]. 3. Let f: ...

### Uniform Continuity - Indian Institute of Technology Kanpur

1 Uniform Continuity Let us ﬂrst review the notion of continuity of a function. Let A ‰ IR and f: A ! IR be continuous. Then for each x0 2 A and for given" > 0 ...

### MathCS.org - Real Analysis: 6.2. Continuous Functions

The identity function f(x) = x is continuous in its domain. ... The function f(x) = 1 / x is continuous on ... If f(x) is uniformly continuous in R, and ...

### Continuity and Uniform Continuity - Welcome | Department ...

as required. We show that fis not uniformly continuous on S, i.e. 9">0 8 >0 9x 0 2S9x2S jx x 0j< and 1 x 1 x 0 " : Let "= 1. Choose >0. Let x 0 = min( ;1) and x= x

### Continuity of Functions - UCLA Department of Mathematics

0 + 2 2 = j4(x x 0)(x+ x 0) 3(x x 0)j ... so f 1(x) = x2 42x+2 is continuous. ... We wish to show this function is discontinuous at x 0 = 10.

### Math 312, Intro. to Real Analysis: Homework #6 Solutions

... x0|+1)2+(|x0|+1)|x0|+|x0|2 = 3|x0 ... (f) sin(1/x2) is not uniformly continous on ... 19.5, because it can be extended to a continous function on [0,1] by ...

### real analysis - Uniformly continuous function $\rightarrow ... ...$f$uniformly continuous and$\int_a^\infty f(x) ... We start by finding $x_0 > 1$. Next, we find \$x_1 > x_0 + 2 ... {n\rightarrow \infty } \int_{a}^{b}g_{n}(x ...

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