Continuity and Differentiability
Continuity of Piecewise Functions
JEE Main 2019
Grade 12
Question:
If $f(x) = \begin{cases} a + bx, & x < 1 \\ 4, & x = 1 \\ b - ax, & x > 1 \end{cases}$ is continuous at $x = 1$, then $(a, b)$:
(1) $(1, 3)$
(2) $(3, 1)$
(3) $(0, 4)$
(4) $(2, 2)$
Step-by-Step Solution
Key Concept: Continuity: $\lim_{x \to 1^-} f = a + b = 4$ and $\lim_{x \to 1^+} f = b - a = 4$. Adding: $2b = 8 \Rightarrow b = 4$; subtracting: $2a = 0 \Rightarrow a = 0$. So $(a, b) = (0, 4)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)