Statement I `f(x) = sin x + [x]` is discontinuous at x = 0.
Statement II If g(x) is continuous and f(x) is discontinuous, then g(x) + f(x) will necessarily be discontinuous at x = a.

Options

A. Statement I is correct, Statement II is also correct, Statement II is the correct explanation of Statement I
B. Statement I is correct, Statement II is also correct, Statement II is not the correct explanation of Statement I
C. Statement I is correct, Statement II is incorrect
D. Statement I is incorrect, Statement II is correct.

Answer

To analyze the statements given in the question, we will break down each statement step by step.

### Step 1: Analyze Statement I
We are given the function:
\[ f(x) = \sin x + [x] \]
where \([x]\) is the greatest integer function (also known as the floor function).

#### Step 1.1: Check Continuity at \(x = 0\)
- The greatest integer function \([x]\) is discontinuous at every integer, including \(x = 0\).
- At \(x = 0\):
- \(\sin(0) = 0\)
- \([0] = 0\)
- Thus, \(f(0) = \sin(0) + [0] = 0 + 0 = 0\).

#### Step 1.2: Check Left and Right Limits
- For \(x\) approaching \(0\) from the left (\(x \to 0^-\)):
- \([x] = -1\) (since for values like \(-0.1, -0.5\), the greatest integer is \(-1\)).
- Thus, \(f(0^-) = \sin(0) + [-1] = 0 - 1 = -1\).

- For \(x\) approaching \(0\) from the right (\(x \to 0^+\)):
- \([x] = 0\) (since for values like \(0.1, 0.5\), the greatest integer is \(0\)).
- Thus, \(f(0^+) = \sin(0) + [0] = 0 + 0 = 0\).

#### Step 1.3: Conclusion for Statement I
- Since \(f(0^-) = -1\) and \(f(0^+) = 0\), and \(f(0) = 0\), we find that:
\[ \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x) \]
- Therefore, \(f(x)\) is discontinuous at \(x = 0\).

### Step 2: Analyze Statement II
Statement II states:
"If \(g(x)\) is continuous and \(f(x)\) is discontinuous, then \(g(x) + f(x)\) will necessarily be discontinuous at \(x = a\)."

#### Step 2.1: Understanding Continuity and Discontinuity
- A function \(g(x)\) being continuous means that:
\[ \lim_{x \to a} g(x) = g(a) \]
- If \(f(x)\) is discontinuous at \(x = a\), then:
\[ \lim_{x \to a} f(x) \neq f(a) \]

#### Step 2.2: Check the Sum \(g(x) + f(x)\)
- For the sum \(g(x) + f(x)\):
- We have:
\[ \lim_{x \to a} (g(x) + f(x)) = \lim_{x \to a} g(x) + \lim_{x \to a} f(x) \]
- Since \(g(x)\) is continuous, we can replace \(\lim_{x \to a} g(x)\) with \(g(a)\):
\[ \lim_{x \to a} (g(x) + f(x)) = g(a) + \lim_{x \to a} f(x) \]

#### Step 2.3: Conclusion for Statement II
- Since \(\lim_{x \to a} f(x) \neq f(a)\), it follows that:
\[ g(a) + \lim_{x \to a} f(x) \neq g(a) + f(a) \]
- Therefore, \(g(x) + f(x)\) is also discontinuous at \(x = a\).

### Final Conclusion
- Both statements are true:
- Statement I is true because \(f(x)\) is discontinuous at \(x = 0\).
- Statement II is true because the sum of a continuous function and a discontinuous function is discontinuous.

### Answer
Both Statement I and Statement II are true.