What is Continuity and Differentiability Class 12: Concepts Explained

Understanding Continuity: Definition and Conditions

Continuity of a function at a point means the function is smooth and unbroken there. Formally, a function $f(x)$ is continuous at $x = a$ if these three conditions hold:

• $f(a)$ is defined
• The limit $\,\lim_{x \to a} f(x)$ exists
• $\,\lim_{x \to a} f(x) = f(a)$

If these are true, the function has no gap, jump, or hole at $a$. If a function is continuous at every point in its domain, it is called continuous on that domain.

Example

Consider $f(x) = x^2$. Check continuity at $x=2$:

• $f(2) = 4$ (defined)
• $\,\lim_{x \to 2} x^2 = 4$
• Limit equals function value

Hence, $f(x)$ is continuous at $x=2$.

Continuity is vital in calculus because it ensures the function behaves predictably and can be differentiated.

Differentiability: What Does It Mean in Class 12 Maths?

Differentiability of a function at a point means the function has a well-defined derivative there. In simple terms, the graph has a tangent line without any sharp corners or cusps.

A function $f(x)$ is differentiable at $x = a$ if the derivative

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

exists.

If $f'(a)$ exists, the function is differentiable at $a$. Differentiability implies continuity, but the reverse is not always true.

Example

Check differentiability of $f(x) = |x|$ at $x=0$:

• Left-hand derivative $= \lim_{h \to 0^-} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$
• Right-hand derivative $= \lim_{h \to 0^+} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1$

Since left and right derivatives differ, $f(x)$ is not differentiable at $0$.

Understanding differentiability helps in graph analysis and solving calculus problems.

Relationship Between Continuity and Differentiability

Continuity and differentiability are closely linked but distinct concepts:

Important Notes

• A function can be continuous but not differentiable (e.g., $f(x) = |x|$ at $x=0$).
• Differentiability guarantees continuity but adds the condition of a smooth slope.

This relationship is fundamental in Class 12 NCERT Maths and helps in understanding function behaviour.

Types of Discontinuities and Their Impact on Differentiability

Discontinuities break the smoothness of a function and affect differentiability.

Common Types of Discontinuities:

• Removable Discontinuity: A hole in the graph; limit exists but not equal to function value.
• Jump Discontinuity: Function jumps from one value to another; left and right limits differ.
• Infinite Discontinuity: Function approaches infinity near the point.

Effect on Differentiability:

• At points of discontinuity, a function is not differentiable.
• Differentiability requires the function to be continuous first.

Example

Consider:

$$ f(x) = \begin{cases} 1, & x < 0 \\ 2, & x \geq 0 \end{cases} $$

At $x=0$, there is a jump discontinuity. Hence, $f$ is not continuous or differentiable at $0$.

Understanding these discontinuities helps in graph sketching and calculus problem-solving.

How to Check Continuity and Differentiability: Step-by-Step Method

Follow these steps to check continuity and differentiability at a point $x=a$:

Checking Continuity:

1. Verify $f(a)$ is defined. 2. Calculate left-hand limit $\lim_{x \to a^-} f(x)$. 3. Calculate right-hand limit $\lim_{x \to a^+} f(x)$. 4. If both limits exist and equal $f(a)$, function is continuous at $a$.

Checking Differentiability:

1. Calculate left-hand derivative:

$$ f'_-(a) = \lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h} $$

2. Calculate right-hand derivative:

$$ f'_+(a) = \lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h} $$

3. If $f'_-(a) = f'_+(a)$ and finite, $f$ is differentiable at $a$.

Worked Example

Check continuity and differentiability of

$$ f(x) = \begin{cases} x^2, & x \leq 1 \\ 2x - 1, & x > 1 \end{cases} $$ at $x=1$.

• $f(1) = 1^2 = 1$
• Left limit: $\lim_{x \to 1^-} x^2 = 1$
• Right limit: $\lim_{x \to 1^+} (2x - 1) = 2(1) - 1 = 1$

Function is continuous at $x=1$.

• Left derivative:

$$ f'_-(1) = \lim_{h \to 0^-} \frac{(1+h)^2 - 1}{h} = \lim_{h \to 0^-} \frac{1 + 2h + h^2 - 1}{h} = \lim_{h \to 0^-} (2 + h) = 2 $$

• Right derivative:

$$ f'_+(1) = \lim_{h \to 0^+} \frac{2(1+h) - 1 - 1}{h} = \lim_{h \to 0^+} \frac{2 + 2h - 1 - 1}{h} = \lim_{h \to 0^+} \frac{2h}{h} = 2 $$

Since left and right derivatives are equal, $f$ is differentiable at $x=1$.

Important Formulas and Theorems in Continuity and Differentiability

Here are some key formulas and theorems from Class 12 NCERT Maths:

• Definition of Derivative:

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

• Continuity Condition:

$$ \lim_{x \to a} f(x) = f(a) $$

• Differentiability Implies Continuity:

If $f$ is differentiable at $a$, then $f$ is continuous at $a$.

• Non-Differentiability Cases:

• Sharp corners (e.g., $|x|$ at 0)
• Discontinuities
• Vertical tangents

• Sum, Product, and Quotient Rules:

For differentiable functions $f$ and $g$:

$$ (f+g)' = f' + g' $$

$$ (fg)' = f'g + fg' $$

$$ \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}, \quad g \neq 0 $$

These formulas are essential for solving calculus problems involving continuity and differentiability.