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The problem asks to find the value of \( a \) such that the function \( f(x) \) is continuous at \( x = 1 \).

The function \( f(x) \) is defined as:

\[
f(x) = 
\begin{cases} 
x + 1 & , x \neq 1 \\
a & , x = 1
\end{cases}
\]

To ensure continuity at \( x = 1 \), the following condition must hold:

\[
\lim_{x \to 1} f(x) = f(1)
\]

1. First, calculate \( \lim_{x \to 1} f(x) \). Since \( f(x) = x + 1 \) for \( x \neq 1 \), we have:

\[
\lim_{x \to 1} f(x) = \lim_{x \to 1} (x + 1) = 1 + 1 = 2
\]

2. For continuity at \( x = 1 \), the value of the function at \( x = 1 \) must equal the limit. Therefore:

\[
f(1) = a = 2
\]

Thus, the value of \( a \) that makes the function continuous is \( a = 2 \).

Would you like further details, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you check if a piecewise function is continuous at a specific point?
2. What is the difference between continuity and differentiability of a function at a point?
3. Can a function be continuous but not differentiable at a point? Provide an example.
4. How would you modify the function if it was required to be differentiable at \( x = 1 \)?
5. What happens if the value of \( a \) is not equal to 2 in this function?

**Tip:** Always check both left-hand and right-hand limits when determining the continuity of a piecewise function at a specific point.

Given the function f(x) = {x + 1, x ≠ 1; a, x = 1}, find the value of a such that the function is continuous at x = 1.

Learn how to find the value of 'a' such that the piecewise function f(x) = {x + 1, x ≠ 1; a, x = 1} is continuous at x = 1. This problem involves key concepts of continuity, limits, and piecewise functions. Suitable for students in Grades 9-11.

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