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Relation between Continuity and Differentiability

If f x =left ...

Question

If $f (x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} x - 1 & x < 0 \frac{1}{4}, & x = 0 x^{2}, & x > 0 \end{matrix}$ , then

A

{lim}_{x \to 0 +} f (x) = 1

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B

{lim}_{x \to 0 -} f (x) = 1

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C

f(x) is discontinuous at x = 0

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D

None of these

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Solution

The correct option is C f(x) is discontinuous at x = 0
Clearly from curve drawn of the given function f(x)is discontinuous at x = 0.

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f (x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} x - 1, & x < 0 \frac{1}{4}, & x = 0 x^{2}, & x > 0 \end{matrix}

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f (x) = {\begin{matrix} 1 + x^{2}, & w h e n 0 \leq x \leq 1 1 - x, & w h e n x > 1 \end{matrix}

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f (x) = {\begin{matrix} 2^{\frac{1}{x}}, & f o r x \neq 0, 3, & f o r x = 0 \end{matrix}

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