Question:
If \(f(x) = |x| + |x-1|\), then which one of the following is correct?
यदि \(f(x) = |x| + |x-1|\), तो निम्नलिखित में से कौन सा सही है?
Options:
- \(f(x)\) is continuous at \(x = 0\) and \(x = 1\)
- \(f(x)\) is continuous at \(x = 0\) but not at \(x = 1\)
- \(f(x)\) is continuous at \(x = 1\) but not at \(x = 0\)
- \(f(x)\) is neither continuous at \(x = 0\) nor at \(x = 1\)
- \(f(x)\), \(x = 0\) और \(x = 1\) दोनों पर सतत है
- \(f(x)\), \(x = 0\) पर सतत है लेकिन \(x = 1\) पर नहीं
- \(f(x)\), \(x = 1\) पर सतत है लेकिन \(x = 0\) पर नहीं
- \(f(x)\) न तो \(x = 0\) पर सतत है और न ही \(x = 1\) पर
Correct Answer: a)
Explanation:
The modulus functions \(|x|\) and \(|x-1|\) are continuous for all real numbers \(x \in \mathbb{R}\).
The sum of two continuous functions is always continuous everywhere.
Specifically at \(x = 0\):
\(\lim_{x \to 0} f(x) = |0| + |0 – 1| = 1 = f(0)\).
At \(x = 1\):
\(\lim_{x \to 1} f(x) = |1| + |1 – 1| = 1 = f(1)\).
Thus, \(f(x)\) is continuous at both \(x = 0\) and \(x = 1\).
Hence, the correct option is (a).
मापांक फलन \(|x|\) तथा \(|x-1|\) सभी वास्तविक संख्याओं \(x \in \mathbb{R}\) पर सतत होते हैं।
दो सतत फलनों का योग भी सदैव प्रत्येक बिंदु पर सतत होता है।
\(x = 0\) पर: \(f(0) = 1\) तथा \(\lim_{x \to 0} f(x) = 1\)।
\(x = 1\) पर: \(f(1) = 1\) तथा \(\lim_{x \to 1} f(x) = 1\)।
अतः \(f(x)\), \(x = 0\) और \(x = 1\) दोनों पर सतत है।
अतः सही विकल्प (a) है।