SEASON 1 EP 2 -- CONTINUITY

 S1  EPISODE 2

CONTINUITY

INTRODUCTION:- 

The concept of limits is a fundamental building block of calculus and mathematical analysis. It is used to define continuity, derivatives, and integrals. 

A function is said to be continuous if it can be drawn without picking up the pencil. Otherwise, a function is said to be discontinuous. Similarly, Calculus in  maths , a function f(x) is continuous at x = c, if there is no break in the graph of the given function at the point. (c, f(c))

DEFINITION:-

A function $f(x)\; is\; said\; to\; be\; continuous\; at\; a\; point$$x = c$ if the following three conditions are satisfied:

1. The function is defined at $c$:

   $$f(c) \text{ exists.}$$
   

2. The limit of the function exists as $x$ approaches $c$:

   $$\lim_{x \to c} f(x) \text{ exists.}$$
   

3. The limit equals the function's value:

   $$\lim_{x \to c} f(x) = f(c).$$

Types of Continuity:-

a) Continuity on an Interval

A function $f(x)$ is said to be continuous on an interval if it is continuous at every point within the interval.

• Open Interval ($a, b$): Continuous for all $x \in (a, b)$.
• Closed Interval $[a, b]$: Continuous for all $x\in [a,b],\; with\; special\; consideration:$
  • At $x=\mathrm{a: }$$\lim_{x \to a^+} f(x) = f(a)$ (right-hand limit).
  • At $x=\mathrm{b: }$$\lim_{x \to b^-} f(x) = f(b)$ (left-hand limit).

b) Pointwise Continuity

A function is continuous at a specific point $x = c$.

c) Uniform Continuity

A function $f(x)$ is uniformly continuous on an interval if, for every $ϵ>\mathrm{0,\; there\; exists\; a }$$\delta > 0$ such that for all $x_1, x_2 \in$ the interval:

$$|x_1 - x_2| < \delta \implies |f(x_1) - f(x_2)| < \epsilon$$

Uniform continuity is stronger than ordinary continuity because $\delta$ depends only on $\epsilon$, not on $x_1$ or $x_2$.

Types of Discontinuities:-

a) Removable Discontinuity

Occurs when:

$$\lim_{x \to c} f(x) \text{ exists, but } f(c) \text{ is not defined or } \lim_{x \to c} f(x) \neq f(c).$$

This can be fixed by redefining $f(c)$appropriately.

b) Jump Discontinuity

Occurs when:

$$\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x).$$

The left-hand limit and right-hand limit are unequal.

c) Infinite Discontinuity

Occurs when:

$$\lim_{x \to c^-} f(x) \text{ or } \lim_{x \to c^+} f(x) \text{ is infinite.}$$

d) Oscillatory Discontinuity

Occurs when the function oscillates infinitely as $x$ approaches $c$, and the limit does not exist.

Properties of Continuous Functions:-

1. Algebra of Continuous Functions:

   • If $f(x)$and $g(x)$ are continuous at $x = c$, then:
     • $f(x) + g(x)$and $f(x)-g(x$) are continuous.
     • $f(x) \cdot g(x)$ is continuous.
     • $\frac{f(x)}{g(x)}$ is continuous if $g(c) \neq 0$.

2. Composition Rule:

   • If $f(x)$is continuous at $x = c$and $g(x)\; is\; continuous\; at$$f(c)$, then $g(f(x))$ is continuous at $x = c$.

3. Intermediate Value Theorem:

   • If $f(x)$ is continuous on $[a, b]$ and $k$ is a value between $f(a)$and $f(b)$, then there exists $c\in [a,b]\; such\; that:$$$f(c) = k.$$
     

4. Extreme Value Theorem:

• If $f(x)$ is continuous on a closed interval $[a, b]$, then $f(x)attains\; both\; a\; maximum\; and\; a\; minimum\; value\; on$$[a, b]$.

Examples of Continuous Functions:-

• Polynomial Functions: Continuous everywhere.
• Rational Functions: Continuous wherever the denominator is non-zero.
• Trigonometric Functions: Continuous wherever they are defined.
• Exponential and Logarithmic Functions: Continuous on their respective domains.
• Absolute Value Function: Continuous everywhere.

Applications of Continuity:-

1. Differentiability: A function must be continuous to be differentiable (though the reverse is not true).
2. Integration: Continuous functions are Riemann integrable.
3. Physics: Continuous functions model real-world phenomena without abrupt changes (e.g., motion, temperature).

That's it for this episode , hope you gained knowledge and this blog was helpful for you , do share it with your friends , see you in next episode with a new topic in this series 😊.

KEEP LEARNING 🕮

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THANK YOU 😊

CIA 1(A)

NAME- SWARA ARYA 

REG NO. 24215226

CLASS-2 BDA

REFRENCES:-

• BYJUS
• KHAN ACADEMY
• WIKIPEDIA
• BOOK- Differential calculus for beginners by Joseph Edwards
• GOOGLE

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