Continuity

Check the continuity of the function
f (x) = [tan x] at x = 0.
L.H.L. (at x = 0)
= \rightarrow ^- [tan x] = \rightarrow [tan (0 - h)]
= \rightarrow [tan h] = [tan 0] = [0] = 0
R.H.L. (at x = 0)
= \rightarrow ^+ [tan x] = \rightarrow [tan (0 - h)]
= \rightarrow [tan h] = [tan 0] = [0] = 0
Now, determine the value of f(x) at x = 0.
f (0) = [tan 0] = [0] = 0
Hence, f (x) is continuous at x = 0.

Step 1: Analyze the piecewise function.
The function is continuous at , but to determine if it is a point of minima or maxima, we compute its derivative and check for concavity.
Step 2: Conclusion.
Thus, is a point of minima.
Final Answer:

The function is defined piecewise: 1. for , 2. for . We are asked to find the values of and such that is continuous at . Step 1: Continuity condition at . For to be continuous at , the following condition must hold: This means the left-hand limit (as ) must equal the right-hand limit (as ), and both must equal the value of the function at . Step 2: Left-hand limit. For , . So, the left-hand limit is: Step 3: Right-hand limit. For , . So, the right-hand limit is: Step 4: Equating the limits. For continuity, the left-hand and right-hand limits must be equal. So, we have: Simplifying: Subtracting 1 from both sides: Thus, the relationship between and is: Step 5: Conclusion. Therefore, the correct answer is (c) .

Step 1: Applying continuity conditions.
For the function to be continuous at , the values of and must satisfy the condition . Using this condition, we solve for and .

Step 2: Conclusion.
The values of and are and , corresponding to option (2).

The function is continuous everywhere, as both sine and cosine functions are continuous. Therefore, there are no points of discontinuity.
Final Answer:

Step 1: The function is oscillating as , which makes it non-differentiable at . However, it is continuous at .

Final Answer:

Step 1: The function represents the fractional part of , which has discontinuities at integer values of .
Step 2: In the interval , there are 6 integer points where the function will be discontinuous: .

Final Answer:

The function is differentiable at all points within the given interval. Therefore, the answer is 0 points where the function is not differentiable.
Final Answer:

By evaluating the function and considering the behavior at , the function is not continuous at any point.
Final Answer: