Limits And Derivatives

Step 1: {Check the points of non-differentiability}
We know that for a function to be differentiable at a point, both the left-hand derivative (LHD) and right-hand derivative (RHD) must be equal at that point. At , the left-hand derivative is 0, but the right-hand derivative is . Therefore, is not differentiable at . Similarly, at , the left-hand derivative is , and the right-hand derivative is 0, meaning is not differentiable at . Thus, .
Step 2: {Calculate }
Since , we find:

Step 1: {Check for continuity at }
We need to find the limit of as : Thus, is continuous at .
Step 2: {Check for differentiability at }
We now check if is differentiable at by finding the derivative at this point: Thus, is differentiable at .

To determine the continuity of at , we need to check if the left-hand limit (LHL), the right-hand limit (RHL), and the function's value at are all equal.
Step 1: Calculate the Left-Hand Limit (LHL)
Factor the numerator and denominator: Cancel the common factor : Substitute :
Step 2: Calculate the Right-Hand Limit (RHL)
Factor the numerator and denominator: Cancel the common factor : As approaches 1 from the right (i.e., ), the numerator approaches 2, and the denominator approaches 0 from the positive side. Thus, the limit is .
Step 3: Evaluate the function at
From the definition of , we have .
Step 4: Compare LHL, RHL, and
We have:
Conclusion:
Since LHL RHL, the limit does not exist. Therefore, the function is discontinuous at . Also, although LHL = , the function is still discontinuous at because the RHL is not equal to these values. For a function to be continuous at a point, the LHL, RHL, and the value of the function at that point must all be equal.

Step 1: {Analyze evenness or oddness}
Thus, is an odd function.
Step 2: {Verify increasing nature}
The derivative for all , indicating is strictly increasing over .