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To check the continuity of the function \( f(X) \) at \( x = a \), we need to verify if the following three conditions are met:

1. \( f(a) \) is defined.
2. \(\lim_{x \to a} f(x)\) exists.
3. \(\lim_{x \to a} f(x) = f(a)\).

Let's examine each of these conditions for the given piecewise function:

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

### 1. \( f(a) \) is defined:
From the given function, \( f(a) = 0 \). Therefore, \( f(a) \) is defined.

### 2. \(\lim_{x \to a} f(x)\) exists:
We need to find the limit of \( (x-a)\sin\left(\frac{1}{x-a}\right) \) as \( x \) approaches \( a \).

Let \( t = x - a \). As \( x \to a \), \( t \to 0 \). The expression \( (x-a)\sin\left(\frac{1}{x-a}\right) \) can be rewritten in terms of \( t \):

\[ f(x) = t \sin\left(\frac{1}{t}\right) \]

We now need to find the limit of \( t \sin\left(\frac{1}{t}\right) \) as \( t \to 0 \).

Since \(\sin\left(\frac{1}{t}\right)\) is bounded between -1 and 1, we have:

\[ -|t| \leq t \sin\left(\frac{1}{t}\right) \leq |t| \]

As \( t \to 0 \), \( |t| \to 0 \). By the Squeeze Theorem, we get:

\[ \lim_{t \to 0} t \sin\left(\frac{1}{t}\right) = 0 \]

Thus,

\[ \lim_{x \to a} (x-a)\sin\left(\frac{1}{x-a}\right) = 0 \]

### 3. \(\lim_{x \to a} f(x) = f(a)\):
We found that \(\lim_{x \to a} f(x) = 0 \) and \( f(a) = 0 \).

Therefore,

\[ \lim_{x \to a} f(x) = f(a) \]

Since all three conditions for continuity are met, the function \( f(X) \) is continuous at \( x = a \).

Would you like further details or have any questions?

Here are five related questions you might be interested in:
1. How do you apply the Squeeze Theorem in other contexts?
2. Can you explain the concept of a removable discontinuity?
3. How does differentiability relate to continuity?
4. What are the different types of discontinuities?
5. Can you show examples of continuous but non-differentiable functions?

**Tip:** When working with limits, remember that trigonometric functions like sine and cosine are bounded, which can be very useful when applying the Squeeze Theorem.

Explore how to determine the continuity of the function f(X) using limits and the Squeeze Theorem. This guide covers three essential conditions for continuity, suitable for advanced high school students and college-level calculus.

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