Problem

Consider the function

the biggest integer smaller than or equal to x. What can you say about the continuity of f(x)?

Solution

The graph of f(x) will help you out:

Certainly the function looks continuous along the places where the graph shows a horizontal line. It appears to be discontinuous near the integers, but clearly f is defined for each integer, so we should see what happens there.

For instance, consider x = 1. What are the limits of the function as x approaches 1?

since x is always greater than 1 when we take this limit.

since x is always less than 1 when we take this limit.

So we can see that f(x) is not continuous at x = 1. In general, if k is any integer,

So f(x) is continuous at all points except the integers.