Say you have the function f(x) = (x²−1)/(x−1). Substitute x=1, and the denominator becomes 0, so f(1) is completely undefined. But at every point other than x=1, this expression is exactly identical to (x−1)(x+1)/(x−1) = x+1. So move x very, very close to 1 (0.999, 1.001, 1.0001...), and f(x) gets infinitely close to 2. This — "the value the function heads toward as x approaches some value" — is called a limit.
The important point here is that the function's value and its limit are two different things. f(1) is undefined (there's no function value), but lim(x→1)f(x)=2 clearly exists (there is a limit value). "The value when you actually arrive at that point" and "the value the trend is heading toward as you approach that point" are different questions, and a limit deals with the latter.
Now you can understand why the concept of continuity requires all three conditions. For a function f to be continuous at x=c, ① f(c) must be defined, ② the limit lim(x→c)f(x) must exist, and ③ the two must be equal (f(c) = the limit value). The example above breaks condition ① (because f(1) doesn't exist), making it discontinuous. This "hole-type" discontinuity is the weakest form of discontinuity — adding just one definition, f(1)=2, makes it continuous right away.
There are more "serious" types of discontinuity than this. A jump discontinuity, where the left-hand limit and right-hand limit head toward different values (like a tiered pricing plan), doesn't even have a limit to begin with (condition ② is broken). An infinite discontinuity, where the function's value grows or shrinks without bound near a certain point — like f(x)=1/x — is likewise a case where the limit doesn't exist. No matter how many definitions you add, these cases can't be fixed into continuity, because the graph itself is fundamentally broken apart.
A limit isn't just a calculation technique — it's the foundation holding up all of calculus, differentiation and integration alike. Differentiation (the instantaneous rate of change), which you learn later, is actually defined as "the limit of squeezing the interval of the average rate of change infinitely narrow," and integration (area) is defined as "the limit of cutting rectangles into infinitely many, infinitely thin pieces." Without the concept of a limit, neither differentiation nor integration could even be defined in the first place.
When studying this with kids, it helps to approach it one step at a time, like climbing stairs — write x values in a table as 1, 1.1, 1.01, 1.001, and have them calculate directly what value f(x) approaches. Seeing it confirmed visually with numbers makes the feel of "getting infinitely close" click much better. On our activity page, you can move x yourself with a slider and watch f(x) approach 2, then compare three types of discontinuity — holes, jumps, and divergence — on a graph.