Exercise 12.2 | Continuous & Discontinuous Functions | New 11th Class Math 2025

 Exercise 12.2 | Continuous & Discontinuous Functions | New 11th Class Math 2025

Introduction

In mathematics, one of the most important concepts is the idea of continuity. When we study real functions, we often want to know whether the graph of a function is smooth without breaks, jumps, or gaps. This smoothness is described by the term continuity. On the other hand, if a function has a break, a hole, or a jump, then the function is said to be discontinuous at that point.

In this chapter, students of the New First Year Math 2025 syllabus will learn the formal definition of continuity, different types of discontinuity, and how to test whether a function is continuous at a particular point or not. Exercise 12.2 focuses on these fundamental ideas, providing examples and questions that make the topic clear and easy to understand.

Continuous and Discontinuous Functions

Continuous Function

A function f(x) is called continuous at a point x = a if:

1. f(a) is defined.

2. The limit of f(x) as x → a exists

3. The value of the limit and the value of the function are equal.

In symbols:

f(x) is continuous at x = a if:

lim (x → a) f(x) = f(a)

This means that the function does not break at x = a. The graph of the function passes smoothly through that point.

Example:

f(x) = 2x + 3

This is a straight-line function. At every value of x, the graph is smooth without breaks, so the function is continuous everywhere on the real line.

Discontinuous Function

A function is discontinuous at a point x = a if any of the following conditions fail:

f(a) is not defined.

The limit of f(x) as x → a does not exist.

The limit exists, but it is not equal to f(a).

In such cases, the graph of the function will show a jump, a hole, or a gap at x = a.

Example:

f(x) = 1/x

At x = 0, the function is not defined. Therefore, the function is discontinuous at x = 0.

One-Sided Limits

To study continuity in more detail, we need to understand the idea of one-sided limits.

Left-hand limit (LHL): This is the value of the function as x approaches a point a from the left side (that is, values smaller than a).

Written as:

lim (x → a⁻) f(x)

Right-hand limit (RHL): This is the value of the function as x approaches a point a from the right side (that is, values greater than a).

Written as:

lim (x → a⁺) f(x)

For the overall limit lim (x → a) f(x) to exist, both LHL and RHL must exist and be equal.

Example of One-Sided Limit

Consider f(x) = |x| / x

For x → 0⁻ (approaching 0 from left side):

f(x) = |x| / x = (−x)/x = −1

So, LHL = −1

For x → 0⁺ (approaching 0 from right side):

f(x) = |x| / x = (x)/x = 1

So, RHL = 1

Here, LHL ≠ RHL, so lim (x → 0) f(x) does not exist. This means the function is discontinuous at x = 0.

Criterion for Existence of Limit of a Function

For a limit of f(x) at x = a to exist, the following condition must be satisfied:

lim (x → a⁻) f(x) = lim (x → a⁺) f(x)

In words: The left-hand limit must equal the right-hand limit.

If this condition is satisfied, then we can say:

lim (x → a) f(x) = common value of LHL and RHL.

Otherwise, the limit does not exist.

Continuity of a Function at a Point

Now that we understand limits, we can clearly define continuity at a point.

A function f(x) is continuous at x = a if:

1. f(a) is defined.

2. lim (x → a) f(x) exists.

3. lim (x → a) f(x) = f(a).

Example

Let f(x) = x²

At x = 2:

1. f(2) = (2)² = 4 → defined.

2. lim (x → 2) f(x) = 2² = 4 → exists.

3. lim (x → 2) f(x) = f(2) → 4 = 4.

Hence, the function is continuous at x = 2.

Example of Discontinuity

Let f(x) = (x² − 4) / (x − 2)

At x = 2:

f(2) is not defined because denominator = 0.

lim (x → 2) f(x) = lim (x → 2) [(x − 2)(x + 2)] / (x − 2) = lim (x → 2) (x + 2) = 4.

Here, limit exists but function is not defined at x = 2.

So, f(x) is discontinuous at x = 2.

Types of Discontinuity

1. Removable Discontinuity:

If limit exists but f(a) is not defined (or not equal to limit). Example: (x² − 1) / (x − 1) at x = 1.

2. Jump Discontinuity:

If LHL ≠ RHL. Example: f(x) = |x|/x at x = 0.

3. Infinite Discontinuity:

If function tends to infinity near a point. Example: f(x) = 1/x at x = 0.

Importance of Continuity

In calculus, differentiation and integration are only possible if functions are continuous.

In physics, continuous functions describe real-world phenomena like motion, growth, and temperature.

Discontinuous functions are used in cases where sudden changes or jumps occur (like electric signals).

Exercise 12.2 solution :

 

1. Check whether functions are continuous or discontinuous.

2. Use one-sided limits to test the existence of limits.

3. Apply the criterion lim (x → a⁻) f(x) = lim (x → a⁺) f(x).

4. Prove continuity of polynomials, trigonometric, and exponential functions.

5. Identify removable and jump discontinuities.

Conclusion

The study of limit and continuity is the foundation of calculus. In Exercise 12.2 of First Year Math (New 2025 Edition), students learn the difference between continuous and discontinuous functions, how to apply left-hand and right-hand limits, and how to check the continuity of a function at a point. These concepts will not only strengthen your understanding of functions but also prepare you for advanced topics like differentiation and integration.

By practicing these problems with simple methods, students will gain confidence in identifying continuity and discontinuity in mathematical functions and applying this knowledge to real-life mathematical modeling.

#MathUniverse #FirstYearMath2025 #Exercise12_2 #LimitAndContinuity #ContinuousFunction #DiscontinuousFunction #OneSidedLimit #LeftHandLimit #RightHandLimit #ContinuityAtAPoint #SimpleMath #EasyMethod #MathsForBeginners #Class11Math