Continuity
We will cover following topics
Continuity Of Functions
Let A⊂R
In general, δ
Uniform Continuity Of Functions
A function f:A→R
A function f
Continuous functions can fail to be uniformly continuous if they are unbounded on a finite domain, such as f(x)=1x
PYQs
Continuity and Uniform Continuity Of Functions
1) Show that if a function f
[2018, 15M]
2) Let f(t)=∫t0[x]dx where [x] denotes the largest integer less than or equal to x.
i) Determine all the real numbers t at which f is differentiable.
ii) Determine all the real numbers t at which f is continuous but not differentiable.
[2017, 15M]
3) Let f:R→R be a continuous function such limx→+∞f(x) and limx→−∞f(x) exist and are finite. Prove that f is uniformly continuous on R.
[2016, 15M]
4) Let
fn(x)={0, if x<1n+1sinπx, if x<1n+1≤x≤1n0, if x>1n
Show that fn(x) converges to a continuous function but not uniformly.
[2012, 12M]
5) Let S=(0,1) and f be defined by f(x)=1x where 0<x≤1 (in R). Is f uniformly continuous on S? Justify your answer.
[2011, 12M]
6) Define the function
f(x)={x2sin1x, if x≠00, if x=0.
Find f′(x). Is f′(x) continuous at x=0? Justify your answer.
[2010, 15M]
7) Let f(x)={|x|2+1, if x<1x2+1, if 1≤x<2−|x|2+1, if 2≤x.
What are the points of discontinuity of f, if any? What are the points where f is not differentiable, if any? Justify your answers.
[2009, 12M]
8) Show that if f:[a,b]→R is a continuous function, then f([a,b])=[c,d] for some real numbers c and d, c≤d.
[2009, 15M]
9) If f:R→R is continuous and f(x+y)=f(x)+f(y), for all x,y∈R, then show that f(x)=xf(1) for all x∈R.
[2008, 12M]
10) Prove that the function f defined by
f(x)={1, when x is rational −1, when x is irrationalis nowhere continuous.
[2006, 12M]
11) Show that the function f(x) defined on R by:
f(x)={x, when x is irrational −x, when x is rationalis continuous only at x=0.
[2004, 12M]