For a function $\displaystyle f(x)$ to be continuous at a given point $\displaystyle (a, f(a))$, it must meet the following two conditions:

1.) The point $\displaystyle (a, f(a))$ must exist, and

2.) $\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$.

Examining the above graph, $\displaystyle f(x)$ is continuous at every possible value of $\displaystyle x$ except for $\displaystyle x=0$. Thus, $\displaystyle f(x)$ is continuous on the interval $\displaystyle (-\infty,0)\cup(0,\infty)$.

For a function $\displaystyle f(x)$ to be continuous at a given point $\displaystyle (a, f(a))$, it must meet the following two conditions:

1.) The point $\displaystyle (a, f(a))$ must exist, and

2.) $\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$.

Examining the above graph, $\displaystyle f(x)$ is continuous at every possible value of $\displaystyle x$ except for $\displaystyle x=-1$ and $\displaystyle x=1$. Thus, $\displaystyle f(x)$ is continuous on the interval $\displaystyle (-\infty,-1)\cup(-1,1)\cup(1,\infty)$.

For a function $\displaystyle f(x)$ to be continuous at a given point $\displaystyle (a, f(a))$, it must meet the following two conditions:

1.) The point $\displaystyle (a, f(a))$ must exist, and

2.) $\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$.

Examining the above graph, $\displaystyle f(x)$ is continuous at every possible value of $\displaystyle x$ except for $\displaystyle x=0$ and $\displaystyle x=1$. Thus, $\displaystyle f(x)$ is continuous on the interval $\displaystyle (-\infty,0)\cup(0,1)\cup(1,\infty)$.

For a function to be continuous at a given point , it must meet the following two conditions:

1.) The point must exist, and

2.) .

Examining the above graph, is continuous at every possible value of except for . Thus, is continuous on the interval .

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

Examining the above graph,  is continuous at every possible value of  except for  and $\displaystyle x=2$. Thus,  is continuous on the interval $\displaystyle (-\infty,0)\cup(0,2)\cup(2,\infty)$.

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

Examining the above graph,  is continuous at every possible value of  except for $\displaystyle x=-1$. Thus,  is continuous on the interval $\displaystyle (-\infty,-1)\cup(-1,\infty)$.

For a function to be continuous at a given point , it must meet the following two conditions:

1.) The point must exist, and

2.) .

Examining the above graph, is continuous at every possible value of except for $\displaystyle x=11$. Thus, is continuous on the interval $\displaystyle (-\infty,11)\cup(11,\infty)$.

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

Examining the above graph,  is continuous at every possible value of  except for $\displaystyle x=2$. Thus,  is continuous on the interval $\displaystyle (-\infty,2)\cup(2,\infty)$.

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

Examining the above graph,  is continuous at every possible value of  except for $\displaystyle x=5$. Thus,  is continuous on the interval $\displaystyle (-\infty,5)\cup(5,\infty)$.

For a function to be continuous at a given point , it must meet the following two conditions:

1.) The point must exist, and

2.) .

Examining the above graph, is continuous at every possible value of except for $\displaystyle x=-1$, , and $\displaystyle x=1$. Thus, is continuous on the interval $\displaystyle (-\infty,-1)\cup\(-1,0)\cup(0,1)\cup(1,\infty)$.