Unfortunately, I can't show a picture here, but I like one example in the introduction of Wolsey's book, "Integer Programming." It provides a short example of why rounding the answer from a continuous (LP) relaxation of an integer programming problem is not sufficient or even close to the integer optimal solution.

  Consider the integer program

  max 1.00 x1 + 0.64 x2
  50 x1 + 31 x2 <= 250
  3x1 - 2x2 >= -4
  x1, x2 >=0 and integer

  As we can see from Figure 1.1, the linear programming solution (376/193,950/193) is a long way from the optimal integer solution (5,0).

Personally, I like this example because it demonstrates that we need to be careful when working with integer optimization problems. It's not that rounding sometimes doesn't work. Hell, we use continuous algorithms like simplex when solving for integer solutions using things like branch and bound. However, it's not the same and depending on the application it can be dangerous to assume so. This also has implications for things like machine learning because they do use optimization algorithms to find their solutions and they often are based on continuous formulations used to give discrete results. It might work, but we should be really careful when it matters most.