Unit 2: Continuity – सीमान्त मान र निरन्तरता
Limits, Continuity & Discontinuity with Graphical Explanations
Introduction to Continuity
Unit 2: Continuity explores the fundamental concepts of limits and continuity in functions. This unit is crucial for understanding calculus and advanced mathematical analysis. Continuity determines whether a function has breaks, jumps, or holes in its graph.
1. निरन्तरता (CONTINUITY)
Continuity in Graph (लेखाचित्रमा निरन्तरता)
यदि फलनको लेखाचित्र खिच्दा कलम नउचालीकन लगातार खिच्न सकिन्छ भने यसलाई निरन्तरता भनिन्छ ।
If the graph of a function can be drawn without lifting the pencil from the paper, then it is called continuity.
Example:
$f(x) = x^2$, $f(x) = \sin x$, $f(x) = e^x$ are continuous functions.
Mathematical Definition (गणितीय परिभाषा)
यदि कुनै बिन्दुमा परिभाषित फलनको मान र सीमान्त मान एक आपसमा बराबर हुन्छन् भने उक्त बिन्दुमा फलन निरन्तर (Continuous) छ भनी लेख्न सकिन्छ ।
If the functional value at a point and the limit value are equal to each other, then the function is continuous at that point.
$\lim_{x \to a} f(x) = f(a)$
Then $f(x)$ is continuous at $x = a$
Three Conditions for Continuity (निरन्तरताका तीन अवस्थाहरू)
Function must be defined at $x = a$
$f(a)$ को अस्तित्व हुनुपर्छ
Limit must exist at $x = a$
$\lim_{x \to a} f(x)$ को अस्तित्व हुनुपर्छ
Limit must equal function value
$\lim_{x \to a} f(x) = f(a)$ हुनुपर्छ
2. विच्छिन्नता (DISCONTINUITY)
Discontinuity (विच्छिन्नता)
यदि कुनै फलनको वक्र कुनै निश्चित बिन्दुमा गएर छुटेको (Break) छ भने उक्त वक्र दिइएको निश्चित बिन्दुमा विच्छिन्न (Discontinue) भएको मानिन्छ । यस्तो अवस्थामा Gap, hole, cusp र break देखा पर्ने गरी टुटेको देखिन्छन् ।
If the graph of any function has a ‘jump’, hole, gap, cusp, or break at a point, then the function is said to be discontinuous at that point.
Example:
$f(x) = \frac{1}{x}$ has discontinuity at $x = 0$
$f(x) = \tan x$ has discontinuity at $x = \frac{\pi}{2}, \frac{3\pi}{2}, …$
Types of Discontinuity (विच्छिन्नताका प्रकारहरू)
| Type (प्रकार) | Description (विवरण) | Example (उदाहरण) |
|---|---|---|
| Removable Discontinuity (हटाउन सकिने विच्छिन्नता) | Hole in the graph, limit exists but $f(a)$ is undefined or different | $f(x) = \frac{x^2-1}{x-1}$ at $x=1$ |
| Jump Discontinuity (उफ्रने विच्छिन्नता) | Left and right limits exist but are not equal | $f(x) = \begin{cases} 1 & x \geq 0 \\ 0 & x < 0 \end{cases}$ at $x=0$ |
| Infinite Discontinuity (अनन्त विच्छिन्नता) | Function approaches infinity at the point | $f(x) = \frac{1}{x}$ at $x=0$ |
| Oscillatory Discontinuity (दोलन गर्ने विच्छिन्नता) | Function oscillates between values near the point | $f(x) = \sin(\frac{1}{x})$ at $x=0$ |
मुख्य बुँदाहरू (Key Points)
1. Polynomial functions are continuous everywhere.
बहुपदीय फलनहरू सबै ठाउँमा निरन्तर हुन्छन् ।
2. Rational functions are continuous except where denominator is zero.
भिन्नात्मक फलनहरू हर शून्य हुने ठाउँ बाहेक निरन्तर हुन्छन् ।
3. Absolute value functions are continuous everywhere.
निरपेक्ष मान फलनहरू सबै ठाउँमा निरन्तर हुन्छन् ।
4. Trigonometric functions have discontinuities at certain points.
त्रिकोणमितीय फलनहरूका केही बिन्दुहरूमा विच्छिन्नता हुन्छ ।
3. सीमान्त मान (LIMITS)
Left Hand Limit (बायाँ पक्षबाट सीमान्त मान)
फलन $f(x)$ मा $x$ बायाँबाट $a$ को नजिक पुग्छ, त्यसलाई $x \to a^{-}$ अथवा $x \to a-0$ लेख्ने गरिन्छ र $f(x)$ का लागि बायाँबाट $a$ मा सीमान्त मानलाई $\lim_{x \to a^{-}} f(x)$ अथवा $\lim_{x \to a-0} f(x)$ द्वारा जनाइन्छ ।
In a function $f(x)$, as inputs $x$ approaches ‘$a$’ from the left, it is written as $x \to a^{-}$ or $x \to a-0$, and the left hand limit of $f(x)$ is written as $\lim_{x \to a^{-}} f(x)$ or $\lim_{x \to a-0} f(x)$.
Right Hand Limit (दायाँ पक्षबाट सीमान्त मान)
फलन $f(x)$ मा $x$ दायाँबाट $a$ को नजिक पुग्छ, त्यसलाई $x \to a^{+}$ अथवा $x \to a+0$ लेख्ने गरिन्छ र $f(x)$ का लागि दायाँबाट $a$ मा सीमान्त मानलाई $\lim_{x \to a^{+}} f(x)$ अथवा $\lim_{x \to a+0} f(x)$ द्वारा जनाइन्छ ।
In a function $f(x)$, as inputs $x$ approaches ‘$a$’ from the right, it is written as $x \to a^{+}$ or $x \to a+0$, and the right hand limit of $f(x)$ is written as $\lim_{x \to a^{+}} f(x)$ or $\lim_{x \to a+0} f(x)$.
Limit Value (सीमान्त मान)
कुनै बिन्दु $x=a$ मा $f(x)$ का लागि बायाँबाट परिभाषित सीमान्त मान र दायाँबाट परिभाषित सीमान्त मान बराबर हुन्छन् भने यस्तो अवस्थामा बिन्दु $a$ मा $f(x)$ को सीमान्त मान परिभाषित भएको मानिन्छ । यसलाई $\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = \lim_{x \to a} f(x)$ द्वारा लेखिन्छ ।
For the function $f(x)$, if the left hand limit and right hand limit at a point $x=a$ are equal, then the limit value of $f(x)$ at a point $a$ is defined. It is written as $\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = \lim_{x \to a} f(x)$.
Limit Properties (सीमान्त मानका गुणहरू)
| Property (गुण) | Formula (सूत्र) |
|---|---|
| Sum Rule | $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$ |
| Difference Rule | $\lim_{x \to a} [f(x) – g(x)] = \lim_{x \to a} f(x) – \lim_{x \to a} g(x)$ |
| Product Rule | $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$ |
| Quotient Rule | $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \lim_{x \to a} g(x) \neq 0$ |
| Constant Multiple | $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$ |
| Power Rule | $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$ |
Special Limits (विशेष सीमान्त मानहरू)
$\lim_{x \to 0} \frac{\sin x}{x} = 1$
Trigonometric limit
$\lim_{x \to 0} \frac{1 – \cos x}{x} = 0$
Cosine limit
$\lim_{x \to 0} \frac{e^x – 1}{x} = 1$
Exponential limit
$\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$
Logarithmic limit
$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$
Euler’s number limit
4. Graphical Analysis (लेखाचित्र विश्लेषण)
Continuous Function (निरन्तर फलन)
A function that can be drawn without lifting the pencil from paper.
Characteristics: Smooth curve, no breaks, no holes, no jumps. Every point has a defined value.
Jump Discontinuity (उफ्रने विच्छिन्नता)
Removable Discontinuity (हटाउन सकिने विच्छिन्नता)
Infinite Discontinuity (अनन्त विच्छिन्नता)
Understanding Limits Graphically (लेखाचित्रबाट सीमान्त मान बुझ्ने)
Approaching from left
$x \to a^-$ means $x$ approaches $a$ from values less than $a$
Approaching from right
$x \to a^+$ means $x$ approaches $a$ from values greater than $a$
Two-sided limit
$\lim_{x \to a} f(x)$ exists only if left and right limits are equal
5. Full Chapter PDF Manual
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