List of top Mathematics Questions on Continuity asked in MET

Number of points where \(f(x)=[\sin x + \cos x]\) is not continuous in \((0,2\pi)\) is:

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

\( f(x)= \begin{cases} \frac{\sin^3(\sqrt{3}) \cdot \log(1+3x)}{(\tan^{-1}\sqrt{x})^2 (e^{5\sqrt{x}}-1)x}, & x\neq 0 \\ a, & x=0 \end{cases} \) is continuous in \( [0,1] \), then \( a \) equals to

The slope of the tangent at \((x,y)\) to a curve passing through \(\left(1, \frac{\pi}{4}\right)\) is given by \(\frac{y}{x} - \cos^2\left(\frac{y}{x}\right)\) then the equation of the curve is

The value of \( f(0) \), so that the function \( f(x) = \frac{2 - (256 - 7x)^{1/8}}{(5x + 3x)^{1/5} - 2} \), \( x \neq 0 \), is continuous everywhere, is given by

If \( f: \mathbb{R} \to \mathbb{R} \) is defined by \[ f(x) = \begin{cases} \dfrac{2 \sin x - \sin 2x}{2x \cos x}, & \text{if } x \ne 0 \\ a, & \text{if } x = 0 \end{cases} \] then the value of \( a \) so that \( f \) is continuous at \( 0 \) is