Is the function $f$ defined by $f\left(x\right)=\left|x\right|$ continuous at$x=0$.

Is the function $f$ defined by $f\left(x\right)=\left|x-1\right|$ continuous at$x=1$.

Is the function $f$ defined by $f\left(x\right)=x-\left|x\right|$ continuous at$x=0$.

Examine the following function for continuity at the indicated points:

$f\left(x\right)=\left\{\begin{array}{l} x^2 , x\ge 0\\ -x , x<0\end{array}\right. \mathrm{at} x=0$

Examine the following function for continuity at the indicated points:

$f\left(x\right)=\left\{\begin{array}{l} 5x-4 ,if x<1\\ 4x^2-3x ,if x\ge 1\end{array}\right. \mathrm{at} x=1$

Is the function $f$ defined by:

$f\left(x\right)=\left\{\begin{array}{l} 2x^2-3 , x\le 1\\ 5 , x>1\end{array}\right.$

continuous at $x=0$? What about its continuity at $x=1 \mathrm{and} x=2$ ? 

Examine the following function for continuity:

$f\left(x\right)=\left\{\begin{array}{l}\frac{x}{\left|x\right|} , x\ne 0\\ 0 , x=0\end{array}\right. \mathrm{at} x=0$.

Examine the following function for continuity:

$f\left(x\right)=\left\{\begin{array}{l}\frac{x-4}{2\left|x-4\right|} , x\ne 4\\ 0 , x=4\end{array}\right. \mathrm{at} x=4$.

Examine the following function for continuity:

$f\left(x\right)=\left\{\begin{array}{l}\frac{x}{\sin 3x} , x\ne 0\\ 3 , x=0\end{array}\right. \mathrm{at} x=0$.