Discuss the continuity of `f(x) ={:{(cos""(1)/(x)", "x ne 0),(" "1", " x= 0):},`

Answer

To discuss the continuity of the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} \cos\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \]

we will follow these steps:

### Step 1: Identify the points of interest
We need to check the continuity of \( f(x) \) at \( x = 0 \) since the function is defined differently at this point compared to other values of \( x \).

**Hint:** Continuity at a point requires that the limit of the function as \( x \) approaches that point equals the function's value at that point.

### Step 2: Calculate the limit as \( x \) approaches 0
We need to find:

\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \cos\left(\frac{1}{x}\right) \]

As \( x \) approaches 0, \( \frac{1}{x} \) approaches infinity. Therefore, we can rewrite the limit:

\[ \lim_{x \to 0} \cos\left(\frac{1}{x}\right) = \lim_{t \to \infty} \cos(t) \quad \text{(where } t = \frac{1}{x}\text{)} \]

**Hint:** The cosine function oscillates between -1 and 1 as its argument approaches infinity.

### Step 3: Analyze the limit
The limit \( \lim_{t \to \infty} \cos(t) \) does not exist because the cosine function does not settle at any particular value; it keeps oscillating.

**Hint:** If a limit does not exist, it cannot equal a specific value.

### Step 4: Compare the limit with the function value at \( x = 0 \)
We know that:

\[ f(0) = 1 \]

Since the limit as \( x \) approaches 0 does not exist, we conclude that:

\[ \lim_{x \to 0} f(x) \neq f(0) \]

**Hint:** For continuity, we need \( \lim_{x \to a} f(x) = f(a) \). Here, it fails.

### Step 5: Conclusion about continuity
Since the limit does not exist and does not equal \( f(0) \), we conclude that the function \( f(x) \) is not continuous at \( x = 0 \).

**Final Statement:** The function \( f(x) \) is not continuous at \( x = 0 \).

### Summary of Steps:
1. Identify points of interest (check continuity at \( x = 0 \)).
2. Calculate the limit as \( x \) approaches 0.
3. Analyze the limit (it does not exist).
4. Compare the limit with the function value at \( x = 0 \).
5. Conclude about continuity.