Differentiability

The correct option is(D): none of these.
To find the limit of the function f(x)=[x]sin(x)​ as x approaches 0, let's analyze the behavior of the function as x approaches 0 from both the left and the right sides.

As x approaches 0 from the right side (x>0), the greatest integer function [x] is always 0, since all values of x between 0 and 1 are rounded down to 0. Therefore, the function becomes f(x)=0sin(x)​, which is undefined.

As x approaches 0 from the left side (x<0), the greatest integer function [x] is again 0, for the same reason as above. So, the function becomes f(x)=0sin(x)​, which is still undefined.

Since the function is undefined as x approaches 0 from both sides, the limit of f(x) as x approaches 0 does not exist. Therefore, the correct answer is "none of these."

In mathematical terms, this can be formally expressed as:

limx→0​f(x)=undefined.

Answer (a) continuous at all pointsAnswer (c) differentiable at all points except at x = 1 and x = - 1.

Let's consider the function f(x)=max{(1−x),(1+x),2}, which can be defined as follows:

f(x)=⎩⎨⎧​1−x,2,1+x,​if x≤−1if −1≤x≤1if x≥1​

Hence, we can observe that:

• As x approaches -1 from the left (denoted as x→−1−), f(x) approaches 1−(−1)=21−(−1)=2.
• As x approaches -1 from the right (denoted as x→−1+), f(x) approaches 22.
• At x=−1, f(x)=2.

Thus, the left-hand limit, right-hand limit, and the value of f(x) at x=−1 are all equal, implying that f(x) is continuous at x=−1.

It's also evident that f(x) is continuous at x=1 as well.

Additionally, f(x) can be analyzed based on its piecewise structure: it is a polynomial function for x≤−1 and x≥1, and a constant function for −−1≤x≤1. Consequently, f(x) is continuous for all x.

Now, considering the derivatives at x=−1 and x=1:

• The left derivative of f(x) at x=−1 (Lf′(x)) is -1.
• The right derivative of f(x) at x=−1 (Rf′(x)) is 0.
• The left derivative of f(x) at x=1 (Lf′(x)) is 0.
• The right derivative of f(x) at x=1 (Rf′(x)) is 1.
• Hence, differentiable at all points except at x = 1 and x = - 1.