Problem 21 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f(x)=|3-x|+(3+x)\), where \((x)\) denotes the least integer greater than or equal to \(x\), then (A) \(f(x)\) is continuous as well as differentiable at \(x=3\) (B) \(f(x)\) is continuous but not differentiable at \(x=3\) (C) \(f(x)\) is differentiable but not continuous at \(x=3\) (D) \(f(x)\) is neither differentiable nor continuous at \(x=3\)

Step-by-Step Solution

Answer

(D) \(f(x)\) is neither differentiable nor continuous at \(x=3\).

Step 1: Understand the Function Components

Let's identify the components of the function. The function \( f(x) \) is given as \( f(x) = |3-x| + (3+x) \). This involves the absolute value function \( |3-x| \) and the ceiling function, represented as \((3+x)\), which rounds up the value inside the parentheses to the nearest integer.

Step 2: Simplify the Function at x = 3

Evaluate each part of the function when \( x = 3 \). For the absolute value part, \( |3-3| = 0 \), and for the ceiling function \((3+3) = 6\) because \(3+3 = 6\) is already an integer. Thus, \( f(3) = 0 + 6 = 6 \).

Step 3: Check Continuity at x = 3

A function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are equal. Calculate \( f(x) \) as \( x \to 3^+ \) and \( x \to 3^- \):- As \( x \to 3^- \), since \( 3-x \to 0^+ \), \( |3-x| = 3-x \), and \((3+x) = 6\). So, \( f(x) = 3-x + 6 \). Thus, \( f(x) \to 6 \).- As \( x \to 3^+ \), \( 3-x \to 0^- \), thus \( |3-x| = x-3 \), and \((3+x) = 7\). Hence, \( f(x) = x-3 + 7 \). Thus, \( f(x) \to 4 \).Since the left and right-hand limits are not equal, \( f(x) \) is not continuous at \( x = 3 \).

Step 4: Check Differentiability at x = 3

A function is differentiable at a point only if it's continuous at that point. Since we already determined that \( f(x) \) is not continuous at \( x = 3 \), it cannot be differentiable either.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental mathematical concept that is often encountered in various problems. It is denoted as \( |x| \) and represents the distance of a number from zero on the number line. This distance is always non-negative. In the context of our exercise, the function \( |3-x| \) tells us how far the value of \( x \) is from \( 3 \).

• If \( x < 3 \), the expression evaluates as \( 3-x \) because the absolute value essentially "flips" negative distances to positive.
• If \( x > 3 \), the expression \( |3-x| \) simplifies to \( x-3 \), also yielding a positive result.

Understanding this operation is crucial for evaluating the limits and continuity of functions involving absolute values, as it changes behavior based on the input relative to the constant.

Ceiling Function

The ceiling function, denoted by \((x)\), is another important concept in mathematics. It gives us the smallest integer greater than or equal to \( x \). Essentially, this function rounds a number up to the nearest integer.

For the function \( (3+x) \), the ceiling operation will evaluate as follows:

• If \( x = 0 \), \( (3+0) = 3 \).
• If \( x = 0.5 \), \( (3 + 0.5) = 4 \) as the ceiling of 3.5 is 4.
• If \( x = 0.99 \), \( (3 + 0.99) = 4 \) because 3.99 rounds up to 4.

The ceiling function plays a crucial role in discontinuities because of its instantaneous step-like jumps even for minute changes in the value of \( x \), which can drastically affect the continuity of the entire function at specific points.

Discontinuity at a Point

Discontinuity refers to a point on the graph of a function where there is a sudden break. This happens when the left-hand limit and right-hand limit at that point do not equal the function’s value. Let's consider our function \( f(x) = |3-x| + (3+x) \).

To find if there is discontinuity at \( x = 3 \), check the limits:

• As \( x \to 3^- \), the function approaches a value based on \( 3-x \).
• As \( x \to 3^+ \), it approaches a different value due to \( x-3 \) and how the ceiling function shifts.

Since the limits do not match each other or the function value at \( x = 3 \), the function is not continuous. This kind of behavior typically indicates discontinuities caused by absolute value and ceiling functions, where inputs cross integer boundaries.

Limit Evaluation

The evaluation of limits is a crucial process in determining the behavior of functions as they approach specific points. It's essential for checking both continuity and differentiability.

For our function, perform a limit evaluation at \( x = 3 \):

• The left-hand limit \( \lim_{x \to 3^-} f(x) \) considers \( 3-x \) and the ceiling of \( (3+x) \) which becomes \( 6 \).
• The right-hand limit \( \lim_{x \to 3^+} f(x) \) uses \( x-3 \) and an increased ceiling value of \( 7 \), leading to a result of \( 4 \).

Since \( 6 eq 4 \), the function does not meet the criteria for continuity at \( x = 3 \). Limit evaluation is thus a powerful tool for ascertaining whether functions exhibit seamless behavior or if abrupt jumps, like in this case, disrupt continuity or differentiability.