Question

Consider the following two statements.

$P:$ There exist functions $f:ℝ\to ℝ,g:ℝ\to ℝ$ such that 𝑓 is continuous at
𝑥 = 1 and 𝑔 is discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1.

$Q:$ There exist functions $f:ℝ\to ℝ,g:ℝ\to ℝ$ such that both 𝑓 and 𝑔 are
discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1.

Which one of the following holds?

1. Both P and Q are true
2. Both P and Q are false
3. P is true but Q is false
4. P is false but Q is true

Answer: Both P and Q are true

Solution :-

For statement ‘P’ :

Take $f\left(x\right)=x-1$ and $g\left(x\right)=\frac{1}{x-1}\mathrm{.}$ So,

Here, we can observe that function $f$ is continuous at $x=1$ and function $g$ is discontinuous at $x=1$ but 𝑔 ∘ 𝑓 is continuous at $x=1.$

Therefore, the statement ‘P’ is true.

Now, For statement ‘Q’ :

Take $f\left(x\right)=g\left(x\right)=\frac{1}{x-1}\mathrm{.}$ So,

Here, we can observe that both functions $f$ and $g$ are discontinuous at $x=1$ but 𝑔 ∘ 𝑓 is continuous at $x=1.$

Therefore, the statement ‘Q’ is also true.

So, the correct answer is option-1. $\to$ Both P and Q are true