Continuity
We will cover following topics
Continuity Of Functions
Let \(A \subset R\). Then, if for each \(x_{0} \in A\) and for given \(\varepsilon>0\), there exists a \(\delta\left(\varepsilon, x_{0}\right)>0\) such that \(x \in A\) and \(\vert x-x_{0}\vert <\delta \rightarrow \vert f(x)-f\left(x_{0}\right)\vert <\varepsilon\).
In general, \(\delta\) depends on \(\epsilon\) and \(x_0\).
Uniform Continuity Of Functions
A function \(f:A \rightarrow R\) where \(A \subset R\), is said to be uniformly continuous on \(A\) if for a given \(\varepsilon>0\), \(\exists \text { } \delta>0\) such that whenever \(x, y \in A\) and \(\vert x-y \vert < \delta\), we have \(\vert f(x)-f(y) \vert < \varepsilon\).
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 \(f(x)= \dfrac{1}{x}\), \(f(x)= \dfrac {1}{x}\) on \((0,1)\), or if their slopes become unbounded on an infinite domain, such as \(f(x)=x^{2}\), \(f(x)=x^{2}\) 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 \(f(t)=\int_{0}^{t}[x] d x\) 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 \(f: R \rightarrow R\) be a continuous function such \(\lim _{x \rightarrow+\infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) exist and are finite. Prove that \(f\) is uniformly continuous on \(\mathbb{R}\).
[2016, 15M]
4) Let
\(f_{n}(x)=\left\{\begin{array}{cl}{0,} & {\text { if } x<\dfrac{1}{n+1}} \\ {\sin \dfrac{\pi}{x},} & {\text { if } x<\dfrac{1}{n+1} \leq x \leq \dfrac{1}{n}} \\ {0,} & {\text { if } x>\dfrac{1}{n}}\end{array}\right.\)
Show that \(f_{n}(x)\) converges to a continuous function but not uniformly.
[2012, 12M]
5) Let \(S=(0, 1)\) and \(f\) be defined by \(f(x)=\dfrac{1}{x}\) where \(0 < x \leq 1\) (in \(R\)). Is \(f\) uniformly continuous on \(S\)? Justify your answer.
[2011, 12M]
6) Define the function
\(f(x)=\left \{ {\begin{array}{ll}x^{2} \sin \dfrac{1}{x}, & \text { if } x \neq 0 \\ {0,} & {\text { if } x=0}\end{array}} \right .\).
Find \(f^{\prime}(x)\). Is \(f^{\prime}(x)\) continuous at \(x=0\)? Justify your answer.
[2010, 15M]
7) Let \(f(x) = \left\{\begin{array}{ll}{\dfrac{\vert x \vert }{2}+1,} & {\text { if } x< 1} \\ {\dfrac{x}{2}+1,} & {\text { if } 1 \leq x< 2} \\ {-\dfrac{\vert x \vert }{2}+1,} & {\text { if } 2 \leq x}\end{array}\right.\).
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 \(f:[a, b] \rightarrow R\) is a continuous function, then \(f([a, b])=[c, d]\) for some real numbers \(c\) and \(d\), \(c \leq d\).
[2009, 15M]
9) If \(f: R \rightarrow R\) is continuous and \(f(x+y)=f(x)+f(y)\), for all \(x, y \in R\), then show that \(f(x)=x f(1)\) for all \(x \in R\).
[2008, 12M]
10) Prove that the function \(f\) defined by
\[f(x)=\left\{\begin{array}{c}{1,} & {\text { when } x \text { is rational }} \\ {-1,} & {\text { when } x \text { is irrational }}\end{array}\right.\]is nowhere continuous.
[2006, 12M]
11) Show that the function \(f(x)\) defined on \(R\) by:
\[f(x)=\left\{\begin{array}{c}{x,} & {\text { when } x \text { is irrational }} \\ {-x,} & {\text { when } x \text { is rational }}\end{array}\right.\]is continuous only at \(x=0\).
[2004, 12M]