Statement-1: If |f(x)|≤|x|for all x∈R then f is continuous at x = 0Statement-2: If f is continuous then |f | is also continuous

Statement-1: If $| f (x) | \leq | x |$ for all $x \in R$ then $|f|$ is continuous at $x = 0$
Statement-2: If f is continuous then $| f |$ is also continuous

A
STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B
STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C
STATEMENT-1 is True, STATEMENT-2 is False
D
STATEMENT-1 is False, STATEMENT-2 is True

Solution:

$| f (x) | \leq | x |$ for all $x \in R \Rightarrow 0 \leq | f (0) | \leq 0$

So, $| f (0) | = 0$ . Also $0 \leq lim_{x \to 0} | f | (x) \leq lim_{x \to 0} | x | = 0 i.e lim_{x \to 0} | f | (x) = 0$

so $| f |$ is continuous. The statement-2 is also true but is not a correct explanation

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