Question

For every integer \[n\]let \[{a_n}\]and \[{b_n}\]be real numbers. Let function \[f:IR \to IR\]be given by
\[f\left( x \right) = \{ \begin{array}{*{20}{l}}
  {{a_n} + \sin \pi x}&{{\text{for }}x\varepsilon \left[ {2n,2n + 1} \right]} \\
  {{b_n} + \cos \pi x}&{{\text{for }}x\varepsilon \left( {2n - 1,2n} \right)}
\end{array}\], for all integers \[n\]. If \[f\] is continuous, then which of the following holds (s) for all \[n\]?
This question has multiple correct options
A. \[{a_{n - 1}} - {b_{n - 1}} = 0\]
B. \[{a_n} - {b_n} = 1\]
C. \[{a_n} - {b_{n + 1}} = 1\]
D. \[{a_{n - 1}} - {b_n} = - 1\]

Answer 1

Hint: In calculus, a function is continuous at x = a if and only if all three of the following conditions are met:
i) The function is defined at x = a; that is, f(a) equals a real number
ii) The limit of the function as x approaches a exists
iii) The limit of the function as x approaches a is equal to the function value at x = a

Complete step by step answer:
First check the continuity at \[x = 2n\]
Left Hand Limit \[ = \mathop {\lim }\limits_{x \to 2{n^ - }} [f(x)]\]
\[ = \mathop {\lim }\limits_{h \to 0} [f(2n - h)]\]
\[ = \mathop {\lim }\limits_{h \to 0} [{b_n} + \cos \pi (2n - h)]\]
\[ = \mathop {\lim }\limits_{h \to 0} [{b_n} + \cos \pi h]\]
\[ = \] \[{b_n} + 1\]
Right Hand Limit \[ = \mathop {\lim }\limits_{x \to 2{n^ + }} [f(x)] = \mathop {\lim }\limits_{h \to 0} [f(2n + h)]\]
\[ = \mathop {\lim }\limits_{h \to 0} [{a_n} + \sin \pi (2n + h)] = {a_n}\]
\[f(2n) = {a_n} + \sin 2\pi n = {a_n}\]
For continuity, \[\mathop {\lim }\limits_{x \to 2{n^ - }} f(x) = \mathop {\lim }\limits_{x \to 2{n^ + }} f(x) = f(2n)\]
So, \[{a_n} = {b_n} + 1 \Rightarrow {a_n} - {b_n} = 1\]
Now, check the continuity at \[x = 2n + 1\]
Left Hand Limit \[ = \mathop {\lim }\limits_{h \to 0} [{a_n} + \sin \pi (2n + 1 - h)] = {a_n}\]
Right Hand Limit \[ = \mathop {\lim }\limits_{h \to 0} [{b_{n + 1}} + \cos (\pi (2n + 1 - h))] = {b_{n + 1}} - 1\]
\[f(2n + 1) = {a_n}\]
For continuity, \[{a_n} = {b_{n + 1}} - 1 \Rightarrow {a_{n - 1}} = {b_n} - 1\]

Hence both option B and D are the right answer.

Note: A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y, the codomain of the function.
An integer is known colloquially as a number that is writable without a fractional part. It is a whole number that can be positive, negative, or zero.