I have a question regarding graphing a function

Question

Find an example for $f(x)$ that is defined on the interval $(-1,5)$ is continuous everywhere except at $x=0,1,2,3,4$ and has no limit at $x=0,3$. I am completely lost any help would be much appreciated.

Answer 1

HINT

• $\frac 1 {x-a}$ is continuos, not defined only for $x=a$ and has not limit for $x\to a$
• $\frac 1 {(x-a)^2}$ is continuos, not defined only for $x=a$ and has limit for $x\to a$

see for example HERE

Answer 2

Here is a graph one such function.

It is defined everywhere on (-1,5) and not defined at the endpoints.

It has a jump discontinuity at (0,3). This means that the limit does not exist at this point.

It has removable discontinuities at the other points, making the function discontinuous at those points, but limits nonetheless exist.