Let f : R $\to$ R be defined as
$$f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.$$
where a, b, c $\in$ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?
Solution
<p>$$f(x) = \left\{ {\matrix{
0 & {x < 0} \cr
{a{e^x} - 1} & {0 \le x < 1} \cr
b & {x = 1} \cr
{b - 1} & {1 < x < 2} \cr
{ - c} & {x \ge 2} \cr
} } \right.$$</p>
<p>To be continuous at x = 0</p>
<p>a $-$ 1 = 0</p>
<p>to be continuous at x = 1</p>
<p>ae $-$ 1 = b = b $-$ 1 $\Rightarrow$ not possible</p>
<p>to be continuous at x = 2</p>
<p>b $-$ 1 = $-$ c $\Rightarrow$ b + c = 1</p>
<p>If a = 1 and b + c = 1 then f(x) is discontinuous at exactly one point.</p>
About this question
Subject: Mathematics · Chapter: Limits, Continuity and Differentiability · Topic: Limits and Standard Results
This question is part of PrepWiser's free JEE Main question bank. 162 more solved questions on Limits, Continuity and Differentiability are available — start with the harder ones if your accuracy is >70%.