Concept of Continuity

Find the points of discontinuity, if any, of the following functions:  \[f\left( x \right) = \begin{cases}- 2 , & \text{ if }& x \leq - 1 \\ 2x , & \text{ if } & - 1 < x < 1 \\ 2 , & \text{ if }  & x \geq 1\end{cases}\]

Find the values of p and q for which

f(x) = `{((1-sin^3x)/(3cos^2x),`

is continuous at x = π/2.

Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}x^{10} - 1, & \text{ if }  x \leq 1 \\ x^2 , & \text{ if } x > 1\end{cases}\]

Prove that `1/2 "cos"^(-1) ((1-"x")/(1+"x")) = "tan"^-1 sqrt"x"`

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(x+1", if"  x>=1),(x^2+1", if"  x < 1):}`

If function f(x) = `{{:((asinx + btanx - 3x)/x^3,",", x ≠ 0),(0,",", x = 0):}` is continuous at x = 0 then (a² + b²) is equal to ______.

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(x/|x|", if"  x<0),(-1", if"  x >= 0):}`

In the following, determine the value of constant involved in the definition so that the given function is continuou: