Problem 44 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)=|x|+[x-1]\), where \([.]\) is greatest integer function, then \(f(x)\) is: (A) continuous at \(x=0\) as well as at \(x=1\) (B) continous at \(x=0\) but not at \(x=1\) (C) continuous at \(x=1\) but not at \(x=0\) (D) neither continuous at \(x=0\) nor at \(x=1\)

Step-by-Step Solution

Answer

(A) continuous at both \(x=0\) and \(x=1\).

Step 1: Analyze the function components

The function is composed of two parts: the absolute value \(|x|\) and the greatest integer function \([x-1]\). \(|x|\) is continuous everywhere. \([x-1]\) is discontinuous wherever \(x-1\) is an integer, i.e., at integer values \(x=n+1\). This part is potentially discontinuous at \(x=1\) and other integers.

Step 2: Check continuity at x=0

At \(x=0\), compute \(f(0) = |0| + [0-1] = 0 - 1 = -1\). Check left-hand and right-hand limits:\ \(\lim_{{x \to 0^-}} f(x) = |-x| + [-x-1]\) becomes \ \( = 0 + [-1] = -1\); \ \(\lim_{{x \to 0^+}} f(x) = |x| + [x-1]\) becomes \(= 0 + [-1] = -1\). Both limits equal \(f(0)\), so \(f(x)\) is continuous at \(x=0\).

Step 3: Check continuity at x=1

At \(x=1\), compute \(f(1) = |1| + [1-1] = 1 + 0 = 1\). Check left-hand and right-hand limits:\ \(\lim_{{x \to 1^-}} f(x) = |x| + [x-1]\) yields \(1 + [0] = 1\); \ \(\lim_{{x \to 1^+}} f(x) = |x| + [x-1]\) becomes \(1 + [0] = 1\). Both limits equal \(f(1)\), so \(f(x)\) is continuous at \(x=1\).

Step 4: Conclusion

Since \(f(x)\) is continuous at both \(x=0\) and \(x=1\), the correct answer is (A) \(f(x)\) is continuous at both points.

Key Concepts

Greatest Integer Function

The greatest integer function, denoted as \([x]\), is a function that returns the largest integer less than or equal to a given number \(x\). This means:

• If \(x\) is an integer, then \([x] = x\).
• If \(x\) is not an integer, then \([x]\) rounds \(x\) down to the nearest whole number.

For example:

• \([3.7] = 3\)
• \([-2.5] = -3\)

This function is inherently discontinuous at every integer point because it has a "step" jump at each integer. For instance, as \(x\) approaches an integer from the left, \([x]\) stays at the lower integer value, but instantaneous step change is seen once crossing that integer, causing a jump in the function's value. Due to its nature, continuity considerations involving the greatest integer function require careful analysis of points where these jumps, or discontinuities, occur.

Absolute Value Function

The absolute value function, expressed as \(|x|\), represents the distance of a number \(x\) from zero on the number line without considering direction. This results in:

• If \(x \ge 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

For example:

• \(|3| = 3\)
• \(|-4| = 4\)

This function is continuous everywhere, meaning it does not exhibit any breaks or jumps at any point on the real number line. The graph of \(|x|\) has a distinctive 'V' shape, with its vertex at the origin \((0, 0)\). This continuous nature plays a significant role when analyzing functions involving both absolute values and other operations, ensuring smooth transitions except at intersections with discontinuous components like the greatest integer function.

Discontinuity Analysis

Discontinuity analysis focuses on identifying where a function does not remain continuous, exhibiting breaks or jumps. In calculus, a function \(f(x)\) is continuous at point \(x = a\) if the following conditions are met:

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

The exercise emphasizes discontinuity by checking these conditions around specific points, such as zero and one, involving functions like the greatest integer function which naturally disrupt continuity at integral arguments. In the given function \(f(x) = |x| + [x-1]\):

• At \(x = 0\) and \(x = 1\), the function is continuous as it satisfies all the above conditions - the limits from both sides match the actual function value.
• The greatest integer function contributes potential discontinuities especially where its argument is an integer (at \(x = n+1\) for \(n\) being an integer in \([x-1]\)).

By employing the absolute value which is continuous everywhere, and analyzing points for possible jumps, effective strategies can be developed to manage functions with mixed continuity profiles.