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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Is the function defined by $\text{f(x)}= \begin{cases}\text{x} + 5,\ \ \text{if x}\leq 1 \\\text{x} - 5,\ \ \text{if x}>1\end{cases}$
a continuous function?
✓

Answer

Here $\text{f(x)}= \begin{cases}\text{x} + 5,\ \ \text{if x}\leq 1 \\\text{x} - 5,\ \ \text{if x}>1\end{cases}$ Function f is defined at all points of the real line. Let c be any real number. Three cases arise: Case I: c < 1 $^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{(x} + 5) = \text{c} + 5$f(c) = c + 5
$\therefore\ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)} = \text{f(c)}$ $\therefore$ f is continuous at all points x < 1. Case II: c > 1 $^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x} - 5) = \text{c} - 5$f(c) = c - 5
$\therefore\ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)} = \text{f(c)}$ $\therefore$ f is continuous at all points x > 1. Case III: c = 1 $^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{1}^{-}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{1}^{-}}\text{(x} + 5) = 1 + 5 = 6$ $^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{1}^{+}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{1}^{+}}\text{(x} - 5) = 1 - 5 = -4$ $\therefore\ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{1}^{-}}\text{f(x)} \neq ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{1}^{+}}\text{f(x)}$ $\therefore$ f is discontinuous at x = 1.

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