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Q)

Find all points of discontinuity of $$f$$, where $$f$$ is defined by $f(x) = \left\{ \begin{array} {1 1} x^{10} -1 ,& \quad\text{ if$ x $$$\leq 1$$}\\ x^2,& \quad \text{if$x$> 1}\\ \end{array} \right.$

$\begin{array}{1 1} \text{Point of discontinuity is x=1} \\ \text{Point of discontinuity is x=-1} \\ \text{Points of discontinuity are all x<1} \\ \text{Points of discontinuity are all x > 1}\end{array}$

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A)
Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
LHL =$\lim\limits_{\large x\to 1^-}f(x)=(x^{10}-1)$
$\qquad\qquad\qquad\;\;=(1-1)$
$\qquad\qquad\qquad\;\;=0$
RHL =$\lim\limits_{\large x\to 1^+}f(x)=(x^2)$
$\qquad\qquad\qquad\;\;=1$
$f(1)=1^{10}-1$
$\;\;\;\;\;\;\;=0$
$\Rightarrow$LHL $\neq$ RHL $\neq$ f(1)
$f$ is not continuous at $x=1$
Step 2:
At $x=c < 1$
$\lim\limits_{\large x\to c}(x^{10}-1)=(c^{10}-1)$
$\qquad\qquad\;\;\;\;\;=f(c)$
At $x=c > 1$
$\lim\limits_{\large x\to c}(x^2)=(c^2)$
$\qquad\quad\;\;=f(c)$
Step 3:
$f$ is continuous at all points $x\in R-\{1\}$
Therefore point of discontinuity is $x=1$