Nat and Vincent were walking home from school one day when Nat sighed. "What's wrong, Nat?" Vincent asked. Nat shrugged, "Today in class, we started to learn about limits and I just don't get them." Vincent chuckled, "Well, Nat, limits are all around you." Nat furrowed her brow and just looked at him.

"Here take a look" Vincent walked Nat a little further away. "See how those two streets come from different ways but still meet?" Nat nodded. "Well, that's the same thing as a function being continuous." Vincent explained.

"Ohhh, so that means they are also differentiable, right?" Nat asked, feeling confident in her answer. "No!" Vincent exclaimed. "For a limit to exist, the limit from the left must equal the limit from the right! The slopes have to match to be differentiable! Limits do not exist where the derivative does not exist: a corner, cusp, vertical tangent, or discontinuity. " Vincent began to explain.

"For a limit to exist, it has to be both differentiable and continuous!!" Vincent said. Nat just nodded, "So, would a piecewise exist?" She asked. Vincent chuckled again, "Yes, if it is both differentiable and continuous. Yet, it may have a jump discontinuity."

"Discontinuities?" Nat asked. "Yeah, there are different types of discontinuities in functions." Vincent stated. Nat looked up at him. "Like what?" She asked again. "There are removable, jump, and infinite discontinuities." Vincent replied. Nat looked confused. "Here, let me explain." He replied.