What is Continuity and Differentiability Class 12: Clear Explanation

Definition of Continuity in Class 12 Mathematics

Continuity of a function at a point means the function has no interruption or jump at that point. Formally, a function $f(x)$ is continuous at $x = a$ if the following three conditions hold:

• $f(a)$ is defined.
• The limit $\, \lim_{x \to a} f(x)$ exists.
• The limit equals the function value: $\lim_{x \to a} f(x) = f(a)$.

If these conditions are true for every point in an interval, the function is continuous on that interval.

Example:

Consider $f(x) = x^2$. At $x=2$:

• $f(2) = 4$.
• $\lim_{x \to 2} x^2 = 4$.
• Both are equal, so $f(x)$ is continuous at $x=2$.

Continuity ensures the graph of $f(x)$ can be drawn without lifting the pen.

Understanding Differentiability: What Does It Mean?

Differentiability means a function has a well-defined derivative at a point, which represents the slope of the tangent line to the curve at that point.

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 $a$.

Example:

For $f(x) = x^2$, the derivative at $x=3$ is

$$ f'(3) = \lim_{h \to 0} \frac{(3+h)^2 - 9}{h} = \lim_{h \to 0} \frac{9 + 6h + h^2 - 9}{h} = \lim_{h \to 0} (6 + h) = 6. $$

So, $f(x)$ is differentiable at $x=3$ with slope 6.

Differentiability implies smoothness; no sharp corners or cusps exist at that point.

Relationship Between Continuity and Differentiability

Continuity and differentiability are closely linked but not identical. Important points:

• If a function is differentiable at $x = a$, it must be continuous at $a$.
• However, a function can be continuous at $a$ but not differentiable there.

Example:

The function $f(x) = |x|$ is continuous everywhere but not differentiable at $x=0$ because it has a sharp corner.

This distinction is crucial in calculus and helps identify points where functions behave irregularly.

Types of Discontinuities in Class 12 NCERT

Discontinuities occur when a function is not continuous at a point. The main types are:

• Removable Discontinuity: Limit exists but $f(a)$ is undefined or different.
• Jump Discontinuity: Left-hand and right-hand limits exist but are not equal.
• Infinite Discontinuity: Limits approach infinity.

Example:

Consider

$$ f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1 \\ 2, & x = 1 \end{cases} $$

At $x=1$, $f(x)$ has a removable discontinuity because

$$ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} (x + 1) = 2, $$

which equals $f(1)$. So, redefining $f(1)$ removes the discontinuity.

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

Follow these steps to check if a function is continuous and differentiable at a point $x = a$:

1. Check continuity:

• Verify $f(a)$ is defined.
• Calculate $\lim_{x \to a} f(x)$.
• Confirm $\lim_{x \to a} f(x) = f(a)$.

2. Check differentiability:

• Compute the derivative limit:

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

• Confirm this limit exists and is finite.

If both conditions hold, the function is continuous and differentiable at $x = a$.

Worked Example:

Check continuity and differentiability of $f(x) = x^3$ at $x=2$.

• $f(2) = 8$.
• $\lim_{x \to 2} x^3 = 8$.
• For derivative,

$$ f'(2) = \lim_{h \to 0} \frac{(2+h)^3 - 8}{h} = \lim_{h \to 0} \frac{8 + 12h + 6h^2 + h^3 - 8}{h} = \lim_{h \to 0} (12 + 6h + h^2) = 12. $$

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

Importance of Continuity and Differentiability in Class 12 Exams

Continuity and differentiability form the backbone of calculus in Class 12 NCERT Mathematics. Understanding these concepts helps students:

• Solve limits and derivative problems efficiently.
• Analyze the behaviour of functions graphically and algebraically.
• Apply concepts in real-life problems involving motion, growth, and rates.
• Prepare for competitive exams like JEE and NEET.

Teachers often ask questions on:

• Proving continuity or differentiability at a point.
• Identifying types of discontinuities.
• Applying derivative formulas.

Mastering this chapter ensures a strong foundation for higher studies in mathematics and science.