Understanding Continuity: Check if Function f(X) is Continuous at x = a

Math Problem Statement

Check the continuity for this function

Solution

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

$f(a)$ is defined.
$\lim_{x \to a} f(x)$ exists.
$\lim_{x \to a} f(x) = f(a)$ .

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

(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.

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Continuity

Formulas

$ f(a) $ is defined
$\lim_{x \to a} f(x)$
$\lim_{x \to a} f(x) = f(a)$

Theorems

Squeeze Theorem

Suitable Grade Level

Grades 11-12

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