Limits, Continuity & Differentiability
Differential Calculus-1
star_batch_jee_advanced_2025
Grade 12
Question:
$f$ is a continuous function in $[a, b]$; $g$ is a continuous function in $[b, c]$. A function $h(x)$ is defined as:
$h(x) = f(x)$ for $x \in [a, b]$
$= g(x)$ for $x \in [b, c]$
if $f(b) = g(b)$, then
h(x) has a removable discontinuity at $x = b$
h(x) may or may not be continuous in $[a, c]$
$h(b^-) = g(b^+)$ and $h(b^+) = f(b^+)$
$h(b^+) = g(b^-)$ and $h(b^-) = f(b^+)$
Step-by-Step Solution
Key Concept: A nonremovable discontinuity persists even when function values match if left and right limits are unequal or nonexistent.
If $f(b) = g(b)$, then $h(x) = f(x) - g(x)$ satisfies $h(b) = 0$. For $h(x)$ to have a nonremovable discontinuity at $x = b$, we require $\text{L.H.L.} \neq \text{R.H.L.}$. Given that $f(b) = g(b)$ (continuity condition met) but the left and right limits of $h$ do not exist or differ, the discontinuity cannot be removed. This occurs when either $g(b^-)$ or $g(b^+)$ do not exist.
Correct Answer: 1,3