Real Analysis & Calculus Revision Guide

Real Analysis Complete Real Analysis & Calculus Revision Guide Continuity • Uniform Continuity • Differentiability • Monotone Functions • Sequences • Limit Points • Topology & Theorems 1. Boundedness Theorem If a function f is continuous on a closed interval [a,b], then it is bounded. There exist real numbers M and m such that: m ≤ f(x) ≤ M for all x ∈ [a,b] Example f(x)=x² on [-2,2] Minimum value = 0 Maximum value = 4 Hence f(x) is bounded. Continuous functions on closed intervals never "blow up" to infinity. 2. Extreme Value Theorem If f is continuous on [a,b], then f attains both: Absolute Maximum Absolute Minimum Example f(x)=x² on [-1,2] Minimum = 0 at x=0 Maximum = 4 at x=2 3. Intermediate Value Theorem (IVT) If f is continuous on [a,b] and k lies between f(a) and f(b), then there exists c∈(a,b) such that: f(c)=k Example f(x)=x³ f(1)=1 and f(2)=8 Since 5 lies between 1 and 8, ...

πŸ“š Python Classes and Objects: A Beginner's Guide

 

“Everything in Python is an object.”
Let’s discover how classes and objects work together to build powerful applications!


πŸš€ What are Classes and Objects?

Classes are blueprints.
Objects are real-world entities created using these blueprints.

πŸ”΅ Diagram:

In Python, everything is an object — integers, strings, lists, even complex numbers!

Example: Complex Numbers as Objects

z = 3 - 5j

Here, z is an instance of the complex class.
Attributes like  and  and methods like  come built-in!


Differences between Class Diagram and Object Diagram:

Aspect

Class Diagram

Object Diagram

Focus

Shows the design or blueprint of the system

Shows real examples (objects) at a certain time

Representation

Shows classes, their properties, and actions

Shows objects, their values, and their connections

Static vs Dynamic

Static — stays the same while designing

Dynamic — changes depending on what's happening

Usage

Used to plan and design how the system should work

Used to see how things actually look while running

 


🧱 Defining Your Own Class

Creating a simple empty class:

Now create objects:

Assign attributes dynamically:

Or for s2:

You can even delete attributes:


πŸ”₯ Adding Methods to Classes

Let's make our class more powerful:

Usage:


🎯 Class Variables vs Object Variables

Aspect

Class Variable

Object Variable

Belongs To

Class itself

Individual Objects

Shared?

Yes

No

Example

name, roll


πŸ› ️ Special Methods: Magic Behind the Scenes

Python classes have special methods that control behavior:

Method

Purpose

Constructor

Pretty print

Developer representation

Operator overloading

Make objects callable

Example:


🧬 Real-World Example: 3D Vectors

Imagine building a physics engine or 3D model:

Use it like this:

πŸ—️ Applications: 

3D graphics, simulations, robotics, physics modeling!


πŸ”’ Public, Protected, and Private Variables

Modifier

Syntax

Access Level

Public

Anywhere

Protected

Within class and subclass

Private

Within class only

πŸ”΅ Flowchart:


🧩 Practice Exercises (With Hints! and solution)

1. Sphere Class

  • Hint:

2. Account Class

  • Hint: Withdraw only if balance ≥ amount!

3. Rationals Class

  • Hint:  methods.

4. Country Class

  • Hint: Calculate density as population/area.


🎯 Best Practices for Classes in Python

Example for clean class design:


Advanced Example: Singleton Class

⚠️ Common Mistakes Beginners Make

Forgetting self in method definitions

Using mutable default arguments

Fix:

Directly accessing private attributes

Instead, use getter/setter methods.


πŸ› ️ Want More? Advanced OOP Patterns Preview

  • πŸ”₯ Factory Pattern: Create objects dynamically without exposing the instantiation logic.
  • 🧠 Observer Pattern: Design systems where objects notify others about state changes.

πŸ‘‰ Coming soon in the next blog part! Stay tuned.


πŸ’¬ Let’s Connect!

I’d love to hear from you. πŸš€


πŸ”” Next Challenge Topic Preview:

" Exploring integers with SageMath

 

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