Let f and g be continuous functions in -a- b- and -b- c- respectively- A function hx is defined as hx-fx- x-a-bgx- x-b-c- If fb-gb- then which of the following is-are CORRECT-

F is continuous in [a, b] ⋯(1) g is continuous in [b, c] ⋯(2) f(b)=g(b) ⋯(3) and nbsp;h(x)=f(x), amp; x∈[a,b) g(x), amp; x∈(b,c] ⋯(4) Limit at x ...

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Byju's Answer

Standard XII

Mathematics

Graph of [x]

Let f and g b...

Question

Let $f$ and $g$ be continuous functions in $[a, b]$ and $[b, c]$ respectively. A function $h (x)$ is defined as $h (x) = {\begin{matrix} f (x), & x \in [a, b) g (x), & x \in (b, c] \end{matrix} .$ If $f (b) = g (b),$ then which of the following is/are CORRECT?

h (x)

has a removable discontinuity at

x = b

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h (x)

is continuous in

[a, c]

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h (b^{-}) = g (b^{+})

and

h (b^{+}) = f (b^{-})

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h (b^{+}) = g (b^{-})

and

h (b^{-}) = f (b^{+})

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Solution

The correct option is C $h (b^{-}) = g (b^{+})$ and $h (b^{+}) = f (b^{-})$
$f$ is continuous in $[a, b] \dots (1)$
$g$ is continuous in $[b, c] \dots (2)$
$f (b) = g (b) \dots (3)$
and $h (x) = {\begin{matrix} f (x), & x \in [a, b) g (x), & x \in (b, c] \end{matrix} \dots (4)$

Limit at $x = b :$
$L . H . L . = lim x \to b^{-} h (x) = lim x \to b^{-} f (x)$
$\Rightarrow L . H . L . = f (b)$
and $R . H . L . = lim x \to b^{+} h (x) = lim x \to b^{+} g (x)$
$\Rightarrow R . H . L . = g (b)$
$\Rightarrow L . H . L . = R . H . L .$ $[From (3)]$
But for $h (x)$ to be continuous,
$L . H . L . = R . H . L . = h (b)$
Since value of $h (b)$ is not known,
$∴ h (x)$ has a removable discontinuity at $x = b .$

From $(1)$ and $(2),$ we have
$f (b^{-}) = f (b)$ and $g (b^{+}) = g (b)$
From $(1), (2)$ and $(4),$ we have
$h (x)$ is continuous in $[a, b) \cup (b, c]$
$∴ h (b^{-}) = f (b^{-}) = f (b) = g (b) = g (b^{+}) = h (b^{+})$

$g (b^{-})$ and $f (b^{+})$ are undefined.
$∴ h (b^{+}) = g (b^{+}) = g (b) = f (b) = f (b^{-})$

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