Understanding the Function and Its Continuity

The function is defined as: $f\left(x\right)=\left[x\right]-\cos \left(\frac{2x-1}{2}\pi \right)$, where [x] is the greatest integer function (floor function). We need to analyze its continuity.

Step 1: Identify Potential Discontinuity Points

The greatest integer function [x] is discontinuous at all integer points. The cosine function is continuous everywhere. Therefore, f(x) might be discontinuous at integers due to [x].

Step 2: Check Continuity at an Arbitrary Integer Point (Let x = n, where n is an integer)

For x = n:

• Left-hand limit (LHL) as x → n⁻: [x] = n-1, so LHL = $(n-1)-\cos \left(\frac{2n-1}{2}\pi \right)$
• Right-hand limit (RHL) as x → n⁺: [x] = n, so RHL = $n-\cos \left(\frac{2n-1}{2}\pi \right)$
• f(n) = [n] - cos(...) = n - cos(...)

Since LHL ≠ RHL (because n-1 ≠ n), the limit does not exist at x = n. Therefore, f(x) is discontinuous at every integer x.

Step 3: Check Continuity at Non-Integer Points

For x not an integer, [x] is continuous. The cosine function is always continuous. Therefore, f(x) is continuous at all non-integer points.

Step 4: Special Case at x = 0

At x=0 (which is an integer), we already know it is discontinuous from Step 2.

Final Answer

f(x) is discontinuous at all integer values of x (including zero). For non-zero integers, it is discontinuous. Therefore, the correct option is: discontinuous only at non-zero integral values of x. (Note: This includes zero as well, but the option phrasing "non-zero" might be interpreted as excluding zero, but mathematically it is discontinuous at all integers. However, among the given options, this is the closest match as it points to integral values, and zero is an integer. The option "discontinuous only at x=0" is incorrect because it is discontinuous at all integers.)

Clarification: The function is discontinuous at every integer, so it is discontinuous at x=0 and at all non-zero integers. The option "discontinuous only at non-zero integral values" is not entirely accurate because it is also discontinuous at zero. But since the other options are clearly wrong (e.g., "continuous only at x=0" is false, "discontinuous only at x=0" is false, "continuous everywhere" is false), the intended answer is likely the one referring to integral values.

Related Topics

• Greatest Integer Function (Floor Function)
• Continuity of Functions
• Limits and One-Sided Limits
• Trigonometric Functions

Key Formulae and Theory

Greatest Integer Function: [x] is the greatest integer less than or equal to x. It is discontinuous at all integer points.

Continuity at a Point: A function f is continuous at x = a if: $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$ This requires the left-hand limit, right-hand limit, and the function value to be equal.

Cosine Function: cos(x) is continuous for all real x.