Continuity

Concept Explanation: Continuity of a function at a point means the function has no breaks, jumps, or holes at that point. Formally, a function \(f(x)\) is continuous at \(x = m\) if the left-hand limit (LHL) and right-hand limit (RHL) at \(m\) exist and are equal, and the function's value at \(m\) equals this limit:

\[ \lim_{x \to m^-} f(x) = \lim_{x \to m^+} f(x) = f(m) \]

Indeterminate forms such as \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \times \infty\), \(\infty - \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\) require further analysis to evaluate limits.

Formula Derivation for Limits

Key limit formulas used in continuity and calculus include:

• \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \)
• \( \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \)
• \( \lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e \)
• \( \lim_{x \to 0} \frac{e^x - 1}{x} = 1 \)

These formulas simplify evaluating limits near zero or other points.

Worked Illustration

Find the value of \(k\) such that the function

\[ f(x) = \begin{cases} \frac{\sin x - \cos x}{4x - \pi}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4} \end{cases} \]

is continuous at \( x = \frac{\pi}{4} \).

Solution:

Continuity requires:

\[ \lim_{x \to \frac{\pi}{4}} f(x) = f\left( \frac{\pi}{4} \right) = k \]

Calculate the limit:

\[ \lim_{x \to \frac{\pi}{4}} \frac{\sin x - \cos x}{4x - \pi} \]

Rewrite numerator using sine difference identity:

\[ \sin x - \cos x = \sqrt{2} \sin \left( x - \frac{\pi}{4} \right) \]

Rewrite denominator:

\[ 4x - \pi = 4 \left( x - \frac{\pi}{4} \right) \]

So limit becomes:

\[ \lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2} \sin \left( x - \frac{\pi}{4} \right)}{4 \left( x - \frac{\pi}{4} \right)} = \frac{\sqrt{2}}{4} \lim_{h \to 0} \frac{\sin h}{h} = \frac{\sqrt{2}}{4} \times 1 = \frac{\sqrt{2}}{4} \]

Therefore, \( k = \frac{\sqrt{2}}{4} = \frac{1}{2 \sqrt{2}} \).

Practice Set

• Level 1 – Easy: Verify continuity of \( f(x) = x^2 \) at \( x = 2 \).
• Level 2 – Moderate: Find \(k\) such that \( f(x) = \begin{cases} x^2 + k, & x < 1 \\ 2x + 1, & x \geq 1 \end{cases} \) is continuous at \( x = 1 \).
• Level 3 – Challenging: Determine if \( f(x) = \frac{\sin x}{x} \) is continuous at \( x = 0 \) by defining \( f(0) \) appropriately.

Answer Key

• Level 1: \( \lim_{x \to 2} x^2 = 4 = f(2) \), so continuous.
• Level 2: \( \lim_{x \to 1^-} f(x) = 1 + k \), \( \lim_{x \to 1^+} f(x) = 3 \), set equal: \(1 + k = 3 \Rightarrow k = 2\).
• Level 3: Define \( f(0) = 1 \) since \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), making \(f\) continuous at 0.

Quick Reference

Concept	Condition
Continuity at \(x=m\)	\( \lim_{x \to m^-} f(x) = \lim_{x \to m^+} f(x) = f(m) \)
Indeterminate Forms	\( \frac{0}{0}, \frac{\infty}{\infty}, 0 \times \infty, \infty - \infty, 1^\infty, 0^0, \infty^0 \)

Glossary

• Limit: The value a function approaches as the input approaches a point.
• Left-hand limit (LHL): Limit as \(x\) approaches from the left.
• Right-hand limit (RHL): Limit as \(x\) approaches from the right.
• Continuous function: Function with no breaks or jumps at a point.
• Indeterminate form: Expression where limit evaluation is not straightforward.

Differentiability

Concept Explanation: Differentiability at a point means the function has a defined derivative (slope) at that point. The derivative represents the instantaneous rate of change of the function.

The derivative of \(f(x)\) at \(x = a\) is defined as:

\[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]

Left-hand derivative (LHD) and right-hand derivative (RHD) are defined as:

\[ f'_-(a) = \lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h}, \quad f'_+(a) = \lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h} \]

Function is differentiable at \(a\) if \( f'_-(a) = f'_+(a) \).

Relation between Continuity and Differentiability:

• If a function is differentiable at a point, it is continuous there.
• If a function is not continuous at a point, it is not differentiable there.
• A function can be continuous but not differentiable at a point (e.g., sharp corners).

Formula Derivation for Standard Derivatives

Some standard derivatives are:

Function	Derivative
\(x^n\)	\(nx^{n-1}\)
\(k\) (constant)	0
\(e^x\)	\(e^x\)
\(\log_e x\)	\(\frac{1}{x}\)
\(\sin x\)	\(\cos x\)
\(\cos x\)	\(-\sin x\)
\(\tan x\)	\(\sec^2 x\)
\(\sin^{-1} x\)	\(\frac{1}{\sqrt{1 - x^2}}\)
\(\cos^{-1} x\)	\(-\frac{1}{\sqrt{1 - x^2}}\)
\(\tan^{-1} x\)	\(\frac{1}{1 + x^2}\)

Rules of Derivatives

Product Rule:

\[ \frac{d}{dx} [uv] = u \frac{dv}{dx} + v \frac{du}{dx} \]

Quotient Rule:

\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Worked Illustration

Differentiate \( y = \tan^{-1} \left( \frac{\cos x - \sin x}{\cos x + \sin x} \right) \).

Solution:

Let \( y = \tan^{-1} \left( \frac{\cos x - \sin x}{\cos x + \sin x} \right) \).

Divide numerator and denominator inside the argument by \( \cos x \):

\[ \frac{1 - \tan x}{1 + \tan x} = \tan \left( \frac{\pi}{4} - x \right) \]

So, \( y = \tan^{-1} [ \tan (\frac{\pi}{4} - x) ] = \frac{\pi}{4} - x \).

Differentiate:

\[ \frac{dy}{dx} = 0 - 1 = -1 \]

Practice Set

• Level 1 – Easy: Find \( \frac{d}{dx} (x^3) \).
• Level 2 – Moderate: Differentiate \( y = \sin x \cos x \) using product rule.
• Level 3 – Challenging: Find \( \frac{dy}{dx} \) if \( y = \tan^{-1} \left( \frac{1 - \tan x}{1 + \tan x} \right) \).

Answer Key

• Level 1: \( \frac{d}{dx} (x^3) = 3x^2 \).
• Level 2: Using product rule, \( \frac{dy}{dx} = \cos x \cos x - \sin x \sin x = \cos^2 x - \sin^2 x \).
• Level 3: Simplify inside inverse tangent as \( \tan (\frac{\pi}{4} - x) \), so derivative is \( -1 \).

Quick Reference

Function	Derivative
\(x^n\)	\(nx^{n-1}\)
\(e^x\)	\(e^x\)
\(\sin x\)	\(\cos x\)
\(\cos x\)	\(-\sin x\)
\(\tan x\)	\(\sec^2 x\)
Product Rule	\( (uv)' = u'v + uv' \)
Quotient Rule	\( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \)

Glossary

• Derivative: Instantaneous rate of change of a function.
• Left-hand derivative (LHD): Derivative approaching from the left.
• Right-hand derivative (RHD): Derivative approaching from the right.
• Differentiable function: Function with equal LHD and RHD at a point.
• Product rule: Rule to differentiate product of two functions.
• Quotient rule: Rule to differentiate quotient of two functions.