Continuity Of Functions

Let $A \subset R$ . Then, if for each $x_{0} \in A$ and for given $ε > 0$ , there exists a $δ (ε, x_{0}) > 0$ such that $x \in A$ and $| x - x_{0} | < δ \to | f (x) - f (x_{0}) | < ε$ .

In general, $δ$ depends on $ϵ$ and $x_{0}$ .

Uniform Continuity Of Functions

A function $f : A \to R$ where $A \subset R$ , is said to be uniformly continuous on $A$ if for a given $ε > 0$ , $\exists δ > 0$ such that whenever $x, y \in A$ 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 $f (x) = \frac{1}{x}$ , $f (x) = \frac{1}{x}$ 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.

Theorem

Let $a < b$ and $f : [a, b] \to R$ be continuous. Then, $f$ is uniformly continuous.

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 $f : R \to R$ be a continuous function such $lim_{x \to + \infty} f (x)$ and $lim_{x \to - \infty} f (x)$ exist and are finite. Prove that $f$ is uniformly continuous on $R$ .

[2016, 15M]

4) Let

$f_{n} (x) = {\begin{array}{cl} 0, & if x < \frac{1}{n + 1} \\ \sin \frac{π}{x}, & if x < \frac{1}{n + 1} \leq x \leq \frac{1}{n} \\ 0, & if x > \frac{1}{n} \end{array}$
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 $f (x) = \frac{1}{x}$ where $0 < x \leq 1$ (in $R$ ). Is $f$ uniformly continuous on $S$ ? Justify your answer.

[2011, 12M]

6) Define the function

$f (x) = {\begin{array}{ll} x^{2} \sin \frac{1}{x}, & if x \neq 0 \\ 0, & if x = 0 \end{array}$ .

Find $f^{'} (x)$ . Is $f^{'} (x)$ continuous at $x = 0$ ? Justify your answer.

[2010, 15M]

7) Let $f (x) = {\begin{cases} \frac{| x |}{2} + 1, & if x < 1 \\ \frac{x}{2} + 1, & if 1 \leq x < 2 \\ - \frac{| x |}{2} + 1, & if 2 \leq x \end{cases}$ .

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 $f : [a, b] \to R$ is a continuous function, then $f ([a, b]) = [c, d]$ for some real numbers $c$ and $d$ , $c \leq d$ .

[2009, 15M]

9) If $f : R \to R$ 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) = {\begin{matrix} 1, & when x is rational \\ - 1, & when x is irrational \end{matrix}

is nowhere continuous.

[2006, 12M]

11) Show that the function $f (x)$ defined on $R$ by:

f (x) = {\begin{matrix} x, & when x is irrational \\ - x, & when x is rational \end{matrix}

is continuous only at $x = 0$ .

[2004, 12M]

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