Some Important Functions Of Real Analysis

In Real Analysis, several classes of functions play a foundational role in understanding the behavior of sequences, limits, continuity, differentiability, and integrability. Below is a list of important functions (with examples and notes on their analytical significance):

🔹 1. $f(x) = |x|$

Property	Value / Status
Domain	$\mathbb{R}$
Range	$[0, \infty)$
Continuity	✅ Everywhere
Differentiability	❌ Not at $x = 0$
Even/Odd	✅ Even
Convexity	✅ Convex
Lipschitz Continuous	✅ Yes (L = 1)

🔹 2. Dirichlet Function

$f(x) = \begin{cases} 1 & When & x \in \mathbb{Q} \\ 0 & When & x \in \mathbb{R} \setminus \mathbb{Q} \end{cases}$

Feature	Value
Domain	$\mathbb{R}$
Range	${0, 1}$
Continuous?	❌ Nowhere continuous
Differentiable?	❌ Nowhere differentiable (since not continuous)
Bounded?	✅ Yes, bounded between 0 and 1
Riemann Integrable?	❌ No
Lebesgue Integrable?	✅ Yes, with integral = 0
Measurable?	✅ Yes

🔹 3. [x]

Property	Value
Domain	$\mathbb{R}$
Range	$\mathbb{Z}$
Type	Step function
Continuity	❌ Discontinuous at every integer
Differentiability	❌ Nowhere differentiable
Boundedness	✅ Bounded on any finite interval
Piecewise Constant	✅ Yes — constant between integers

4. $f(x) = x^2 \sin\left(\dfrac{1}{x}\right)$ for $x \ne 0$, $f(0) = 0$

Property	Status
Domain	$\mathbb{R}$
Range	$\mathbb{R}$ (bounded function)
Continuous at $x=0$	✅ Yes
Differentiable at $x=0$	✅ Yes, with $f'(0) = 0$
$f'(x)$ continuous at $x=0$	❌ No
Differentiable elsewhere	✅ Yes

5. $ f(x) =
\begin{cases}
\frac{1}{q}, & \text{if } x = \frac{p}{q} \in \mathbb{Q} \text{ in lowest terms} \\\
0, & \text{if } x \in \mathbb{R} \setminus \mathbb{Q}
\end{cases} $

Feature	Value / Description
Domain	$\mathbb{R}$
Range	$[0,1]$
Bounded	✅ Yes
Continuous at irrationals	✅ Yes
Discontinuous at rationals	✅ Yes
Riemann integrable	✅ Yes
Integral over $[a,b]$	$0$
Differentiable	❌ Nowhere (not even at irrationals)

6. $\operatorname{sgn}(x) =
\begin{cases}
-1, & \text{if } x < 0 \ \\ \ 0, & \text{if } x = 0 \ \\ \ 1, & \text{if } x > 0
\end{cases}$

Property	Description
Domain	$\mathbb{R}$
Range	${-1, 0, 1}$
Even/Odd	Odd function, since $\operatorname{sgn}(-x) = -\operatorname{sgn}(x)$
Discontinuity	Discontinuous at $x = 0$
Continuous elsewhere	✅ Yes — continuous for all $x \ne 0$
Not differentiable at	$x = 0$
Derivative	0 for $x \ne 0$, but undefined at $x = 0$