First slide

Continuity

Question

Let $f (x) = [x] and g (x) = \{\begin{array}{cc} 0, & x \in Z \\ x^{2}, & x \in R - Z \end{array}$ ([.] represents the greatest integer function.) Then

Moderate

A

$lim_{x \to 1} g (x)$ exists but g(x) is not continuous at x = 1
B

f(x) is not continuous at x = l
C

gof is continuous for all x
D

fog is continuous for all x

Solution

Since $lim_{x \to 1^{-}} g (x) = lim_{x \to 1^{+}} g (x) = 1 and g (1) = 0$
g(x) is not continuous at x = 1 but $lim_{x \to 1} g (x)$ exists.
we have $lim_{x \to 1^{-}} f (x) = lim_{h \to 0} f (1 - h) = lim_{h \to 0} [1 - h] = 0$
and $lim_{x \to 1^{+}} f (x) = lim_{h \to 0} f (1 + h) = lim_{h \to 0} [1 + h] = 1$
So, $lim_{x \to 1} f (x)$ does not exist and, hence, f(x) is not continuous at x = 1.
We have $gof (x) = g (f (x)) = g ([x]) = 0 \forall x \in R$
So, gof is continuous for all x.
We have
$\begin{array}{l} fog (x) = f (g (x)) = \{\begin{array}{cc} f (0), & x \in Z \\ f (x^{2}), & x \in R - Z \end{array} \\ = \{\begin{array}{cc} 0, & x \in Z \\ [x^{2}], & x \in R - Z \end{array} \end{array}$
which is clearly not continuous.

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