List of top Mathematics Questions on Continuity asked in KEAM

Let $f(x) = \begin{cases} \frac{1-\sec^2(\alpha x)}{\alpha x^2}, & \text{for } x \neq 0 \\ -3, & \text{for } x = 0 \end{cases}$ be continuous at $x = 0$. Then the value of $\alpha$ is equal to

If \[ f(x)= \begin{cases} x^2, & \text{if } x\leq 2,\\[4pt] 4x-\alpha, & \text{if } x>2, \end{cases} \] is continuous at \(x=2\), then the value of \(\alpha\) is equal to: 

If the function \[ f(x) = \begin{cases} x^2, & \text{for } x < 4 \\ 5x - k, & \text{for } x \geq 4 \end{cases} \] is continuous at \( x = 4 \), then the value of \( k \) is equal to
If \( f(x) = \frac{1}{2 - x} \) and \( g(x) = \frac{1}{1 - x} \), then the point(s) of discontinuity of the function \( g(f(x)) \) is (are):
The number of positive integers that have at most seven digits and contain only the digits 0 and 9 is