A continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. A function is said to be continuous in a point if the one-sided limits on this points are equal to the actual function value on that point.
Let's analyze the one-sided limits at x = 1.
[tex]\lim_{x\to1^-}f(x)=\lim_{x\to1^+}f(x)=3\ne f(1)=4[/tex]
Since the limits are not equal to the function value, the function is not continuous at x = 1. The next point of possible discontinuity would be at x = 2.
We have for x = 2 the following limits:
[tex]\begin{gathered} \lim_{x\to2^-}f(x)=2=f(2) \\ \lim_{x\to2^+}f(x)=3\ne f(2) \end{gathered}[/tex]
Here, the one-sided limit coming from the left is equal to the function value at x = 2, but the one sided limit coming from the right isn't, therefore, the function is only continuous from the left at x = 2.
Reproducing the same analysis for the other points, we get our answer. Since the function is not defined at x = 0, x = 3, and x = 4, the function can't be continuous on those points.
