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Continuity in an Interval

f is a contin...

Question

$f$ is a continuous function in $[a, b]$ and $g$ is a continuous function $h (x)$ is defined as
$h (x) = {\begin{matrix} f (x), x \in [a, b) g (x), x \in (b, c] \end{matrix}$
Now, if $f (b) = g (b)$ , then

h (x)

may or may not be 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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h (x)

has a removable discontinuity at

x = b

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Solution

The correct option is A $h (x)$ may or may not be continuous in $[a, c]$
At $x = b$
${lim}_{x \to b^{-}} h (x) = {lim}_{x \to b^{-}} f (x) = f (b)$
${lim}_{x \to b^{+}} h (x) = {lim}_{x \to b^{+}} g (x) = g (b)$
Given $f (b) = g (b)$
$\Rightarrow$ LHL $=$ RHL
But for $h (x)$ to be continuous
LHL $=$ RHL $= h (b)$
Since $h (b)$ value is not known it may or may not be continuous.

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Similar questions

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f is a continuous function in [a,b] and g is a continuous function h(x) is defined ash(x)={f(x),x∈[a,b)g(x),x∈(b,c]Now, if f(b)=g(b), then

The correct option is A h(x) may or may not be continuous in [a,c]At x=blimx→b−h(x)=limx→b−f(x)=f(b)limx→b+h(x)=limx→b+g(x)=g(b)Given f(b)=g(b)⇒LHL =RHLBut for h(x) to be continuous LHL=RHL =h(b)Since h(b) value is not known it may or may not be continuous.

$f$ is a continuous function in $[a, b]$ and $g$ is a continuous function $h (x)$ is defined as
$h (x) = {\begin{matrix} f (x), x \in [a, b) g (x), x \in (b, c] \end{matrix}$
Now, if $f (b) = g (b)$ , then