Consider the following statements in respect of f(x) = |x| - 1 1. f(x) is continuous at x = 1. 2. f(x) is differentiable at x = 0. Which of the above statements is/are correct?
1 only
Statement 1: Continuity at \(x=1\).
The function \(f(x)=|x|-1\) is the difference of two continuous functions (\(|x|\), which is continuous for all real \(x\), and the constant \(-1\)). Hence \(f\) is continuous everywhere, in particular at \(x=1\). Also \(f(1)=|1|-1=0\) and \(\lim_{x\to 1}f(x)=0\), so \(\lim_{x\to 1}f(x)=f(1)\). Statement 1 is correct.
Statement 2: Differentiability at \(x=0\).
Using the definition of the derivative at \(x=0\) with \(f(0)=|0|-1=-1\):
\(\lim_{h\to 0}\dfrac{f(0+h)-f(0)}{h}=\lim_{h\to 0}\dfrac{|h|-1-(-1)}{h}=\lim_{h\to 0}\dfrac{|h|}{h}\)
Left-hand limit: \(\lim_{h\to 0^{-}}\dfrac{-h}{h}=-1\). Right-hand limit: \(\lim_{h\to 0^{+}}\dfrac{h}{h}=+1\). Since LHL \(\ne\) RHL, \(f\) is not differentiable at \(x=0\) (the graph has a sharp corner). Statement 2 is incorrect.
Hence only Statement 1 is correct.
Similar Questions
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