Graph of a Function or a Just a Relation?

Question

Does this picture say that the function $f(x)$ has the values $1.3$ and $3.8$ at $a$? How can this be a function if it has two outputs for one input? There has to be a hole at either $(a, 1.3)$ or $(a, 3.8)$ or both and so the limit does not exist. Our teacher had us draw a similar graph.

Answer 1

You're correct in assuming that there is a hole at $(a, 1.3)$ or at $(a, 3.8)$. Limit's don't account for holes, so no matter which point has a hole, the limits both still exist - that's what limits are for: detecting values that aren't there. If the absence of an obviously drawn hole in the picture is what's confusing you, it's probably just a bad picture.

Answer 2

As we can see, your function is defined at $[0,a)\cup (a,10]$.

if both limit were equal to $L $, we could put $f (a)=L $, and the new function will be continuous at $[0,10] $.