Problem 82 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \([x]\) denotes the greatest integer less than or equal to \(x\). If \(f(x)=[x \sin p x]\), then \(f(x)\) is (A) continuous at \(x=0\) (B) continuous in \((-1,0)\) (C) differentiable at \(x=1\) (D) differentiable in \((-1,1)\)

Step-by-Step Solution

Answer

The function is continuous at \(x=0\). Choice (A) is correct.

Step 1: Understanding the Function

The function given is \(f(x) = \lfloor x \sin(px) \rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\). This is a floor function inside a trigonometric function.

Step 2: Analyzing Continuity at x=0

Continuity at a point \(x = a\) requires \(\lim_{{x \to a}} f(x) = f(a)\) and \(f(a)\) to be defined. At \(x = 0\), \(f(0) = \lfloor 0 \sin(0) \rfloor = 0\). The limit, \(\lim_{{x \to 0}} \lfloor x \sin(px) \rfloor = 0\), so \(f(x)\) is continuous at \(x = 0\).

Step 3: Analyzing Continuity in (-1,0)

We examine the continuity in the interval \((-1, 0)\). For \(x < 0\), \(x\sin(px)\) can take any real number value leading to steps/discontinuities in \(\lfloor x\sin(px) \rfloor\). Therefore, \(f(x)\) is not continuous in \((-1, 0)\).

Step 4: Checking Differentiability at x=1

For \(f(x)\) to be differentiable at \(x=1\), it must be continuous at \(x=1\) and must have a defined derivative at \(x=1\). Since \(f(x) = \lfloor x\sin(px) \rfloor\) is not a smooth function due to the floor operation, it lacks differentiability at integer points, including \(x=1\).

Step 5: Checking Differentiability in (-1,1)

The presence of a floor function makes \(f(x)\) piecewise constant and thus non-differentiable in any open interval containing integer values. Consequently, \(f(x)\) is not differentiable in \((-1, 1)\) except possibly right at single non-integer points.

Key Concepts

Continuity

Continuity is a fundamental concept in calculus, describing whether a function behaves predictably as its input value approaches a particular point. More formally, a function \( f(x) \) is said to be continuous at a point \( a \) if three conditions are met:

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

In the context of the given function \( f(x) = \lfloor x \sin(px) \rfloor \), continuity is thoroughly examined at specific points and intervals.
At \( x = 0 \), the function is continuous because both the limit \( \lim_{{x \to 0}} \lfloor x \sin(px) \rfloor \) and the function value \( f(0) = 0 \) align.
However, for the open interval \((-1, 0)\), the behavior of \( f(x) \) becomes more erratic due to the floor function, causing multiple discontinuities as \( x \sin(px) \) can assume a wide range of values resulting in step-like changes in \( \lfloor \cdot \rfloor \). This disrupts the continuity across the interval.

Differentiability

Differentiability is a property of a function that refers to the existence of a derivative at a particular point. For a function to be differentiable at a point \( x = a \), it must be continuous at that point and also have a well-defined tangent, represented by the derivative.
The function \( f(x) = \lfloor x \sin(px) \rfloor \) involves the floor function, which inherently creates sharp corners or jumps. These discontinuities imply the absence of a smooth, continuous tangent line, thus making \( f(x) \) non-differentiable at any integer value, such as \( x = 1 \).
In the range \((-1, 1)\), another complication arises since the floor function makes \( f(x) \) appear piecewise constant, altering its differentiability across the interval. A piecewise constant function is flat in segments, providing no slope to calculate with a derivative, leading to a lack of differentiability throughout most of the interval.

Greatest Integer Function

The greatest integer function, also known as the floor function, assigns the greatest integer less than or equal to a real number \( x \). It is denoted by \( \lfloor x \rfloor \). This function is characterized by jumps or steps at each integer, creating a distinct staircase pattern when graphed.
The function within the exercise, \( f(x) = \lfloor x \sin(px) \rfloor \), exploits this property of the greatest integer function. Wherever there are changes in \( x \sin(px) \), the floor function manifests these as steps in \( f(x) \).
These steps are responsible for the function's behavior regarding continuity and differentiability. As it steps across integer values, these shifts introduce discontinuities and eliminate the possibility of a differentiable function, due to the absence of smooth transitions or defined slopes. Understanding the influence of the floor function is crucial to analyzing problems involving the greatest integer function.