First slide

Continuity

Question

$The function f (x) = [x]^{2} - [x^{2}] (where [y] is the tangent integer \leq y) is discontinuous at$

Moderate

A

all integer
B

All integers except 0 and 1
C

all integer except 0
D

All integers except 1

Solution

$if 0 < x < 1, then 0 < x^{2} < 1$
$\Rightarrow [x] = 0 and \Rightarrow [x^{2}] = 0 ∴ f (x) = 0 for 0 < x < 1$
$if 1 \leq x < \sqrt{2}, then 1 \leq x^{2} < 2$
$\Rightarrow [x] = 1, [x^{2}] = 1 ∴ f (x) = 0 for 1 \leq x < \sqrt{2}$
$∴ f is continuous at x = 1$

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