Continuity
We will cover following topics
Continuity Of Functions
Let A⊂R. Then, if for each x0∈A and for given ε>0, there exists a δ(ε,x0)>0 such that x∈A and |x−x0|<δ→|f(x)−f(x0)|<ε.
In general, δ depends on ϵ and x0.
Uniform Continuity Of Functions
A function f:A→R where A⊂R, is said to be uniformly continuous on A if for a given ε>0, ∃ δ>0 such that whenever x,y∈A and |x−y|<δ, we have |f(x)−f(y)|<ε.
A function f is uniformly continuous if it is possible to guarantee that f(x) and f(y) can be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.
Continuous functions can fail to be uniformly continuous if they are unbounded on a finite domain, such as f(x)=1x, f(x)=1x on (0,1), or if their slopes become unbounded on an infinite domain, such as f(x)=x2, f(x)=x2 on the real line.
PYQs
Continuity and Uniform Continuity Of Functions
1) Show that if a function f defined on the open interval (a,b) of R is convex, then f is continuous. Show by example, if the condition of open interval is dropped, then the convex function need not be continuous.
[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]