TS EAMCET · Maths · Continuity and Differentiability
If \([x]\) is the greatest integer function and \(f(x)= \begin{cases}2[x]-\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{cases}\) is a real valued function, then \(f\) is
- A continuous at \(x=0\)
- B continuous at \(x=1\)
- C left continuous at \(x=0\)
- D right continuous at \(x=1\)
Answer & Solution
Correct Answer
(D) right continuous at \(x=1\)
Step-by-step Solution
Detailed explanation
\(f(1) = 2[1]-\frac{1}{|1|} = 2(1)-1 = 1\) \(\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2[x]-\frac{x}{|x|})\) \( = \lim_{x \to 1^+} (2(1)-\frac{x}{x}) = 2-1 = 1\) \(\text{Since } \lim_{x \to 1^+} f(x) = f(1), f \text{ is right continuous at } x=1.\)
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