Let [t] denote the greatest integer [ ] t and [ _x 0 x[ 4 x ] = A].

Question

Let [t] denote the greatest integer $$ \le $$ t
and $$\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A$$.
Then the function,
f(x) = [x2]sin($$\pi $$x) is discontinuous, when x is
equal to :

Accepted Answer

A = $$\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right]$$

= $$\mathop {\lim }\limits_{x \to 0} x\left( {{4 \over x} - \left\{ {{4 \over x}} \right\}} \right)$$

= $$\mathop {\lim }\limits_{x \to 0} \left( {4 - \left\{ {{4 \over x}} \right\}} \right)$$

= 4

Now, when x = $$\sqrt {A + 1} $$ = $$\sqrt 5 $$, f(x) = [x2]sin($$\pi $$x) is discontinuous at this non integer point.

But at x = 2, 3 and 5, f(x) is continuous.

Solutions · 1