To determine the nature of the function [tex]\( f(x) = \frac{a}{1-x} \)[/tex] at [tex]\( x = 1 \)[/tex], we need to analyze the behavior of the function at that specific point.
1. Understand Division by Zero: The function [tex]\( f(x) \)[/tex] involves a fraction where the denominator is [tex]\( 1 - x \)[/tex]. If [tex]\( x \)[/tex] is such that the denominator becomes zero, the function will be undefined at that point. This is a crucial aspect in determining continuity and existence of the function at any specific point.
2. Evaluate the Denominator at [tex]\( x = 1 \)[/tex]: Substituting [tex]\( x = 1 \)[/tex] into the denominator:
[tex]\[
1 - x = 1 - 1 = 0
\][/tex]
This reveals that the denominator becomes zero at [tex]\( x = 1 \)[/tex].
3. Implication of Division by Zero: Since the denominator is zero when [tex]\( x = 1 \)[/tex], the function [tex]\( f(x) = \frac{a}{1-x} \)[/tex] will be undefined at [tex]\( x = 1 \)[/tex]. This means that the function doesn't produce a finite value at this point.
4. Continuity Analysis: For a function to be continuous at a point, it must:
- Be defined at that point.
- Have a limit as [tex]\( x \)[/tex] approaches that point.
- Have the function's value at that point equal to the limit.
In our case, since the function is not defined at [tex]\( x = 1 \)[/tex], it fails the first condition for continuity.
5. Conclusion: Given that the function [tex]\( f(x) \)[/tex] is not defined at [tex]\( x = 1 \)[/tex], it is correct to say the function is discontinuous at this point. Therefore, the function [tex]\( f(x) = \frac{a}{1-x} \)[/tex] at [tex]\( x = 1 \)[/tex] is discontinuous.
Thus, the correct answer is:
[tex]\[
\boxed{3} \quad (c. \text{discontinuous})
\][/tex]
