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Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability2 Marks
Question
Show that the function $f$ given by $f(x)=\left\{\begin{array}{ll} {x^{3}+3,} & {\text { if } x \neq 0} \\ {1,} & {\text { if } x=0} \end{array}\right.$ is not continuous at $x = 0$.
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Answer

The function is defined at $x = 0$ and $f(0) =1$.
When $x \neq 0,$ the function is given by a polynomial.
Hence,
$\mathop {\lim }\limits_{x \to 0} f(x) = \mathop {\lim }\limits_{x \to 0} (x^3 + 3) = 0^3 + 3 = 3$
Since , the limit of $f$ at $x = 0$ does not coincide with $f(0),$ the function is not continuous at $x = 0$.
It may be noted that
$x = 0$ is the only point of discontinuity for this function.

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