List of top Mathematics Questions on Continuity asked in MET

If \(f(x) = \begin{cases} x, & 0 \le x \le 1 \\ 2x - 1, & 1<x \end{cases}\), then

The function \(f(x) = [x]\cos\left[\frac{2x - 1}{2}\right]\pi\), where \([\,\cdot\,]\) denotes the greatest integer function, is discontinuous at

If \(f_n(x) = e^{f_{n-1}(x)}\) for all \(n \in \mathbb{N}\) and \(f_0(x) = x\) then \(\frac{d}{dx}\{f_n(x)\}\) is equal to

Let \(f(x) = x^p \cos(1/x)\) when \(x \neq 0\) and \(f(x) = 0\), when \(x = 0\). Then \(f(x)\) will be differentiable at \(x = 0\), if

The derivative of \(f(x) = 3|2 + x|\) at the point \(x_0 = -3\) is

Derivative of the function \(f(x) = \log_5(\log_7 x)\), \(x>7\) is

If \(u = x^2 + y^2\) and \(x = s + 3t\), \(y = 2s - t\) then \(\frac{d^2 u}{ds^2}\) is equal to

If \(f(x) = \cot^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)\) and \(g(x) = \cos^{-1}\left(\frac{1 - x^2}{1 + x^2}\right)\), then \(\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}\), \(0<a<1/2\), is

\(\lim_{x \to -2} \frac{\sin^{-1}(x + 2)}{x^2 + 2x}\) is equal to

If the function \[ f(x) = \begin{cases} (\cos x)^{1/x^2}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} \] is continuous at \( x = 0 \), then the value of \( k \) is:

\( \lim_{x \to 0} \frac{\sqrt{1 - \cos(2x)}}{\sqrt{2} \cdot x} \) is