StellarThe Intermediate Value Theorem says that if a function is continuous on the closed interval [a,b], and if N is any number between f(a) and f(b), then there exists some value c in (a,b) such that f(c)=N. In simpler terms, a continuous graph cannot jump over a y-value between its starting and ending heights on an interval. If it starts below a number and ends above it, then it must hit that number somewhere in between.
This theorem does not tell us exactly where the value occurs, and it does not tell us how many times it occurs. It only guarantees existence. The theorem is especially useful for proving that an equation has at least one solution in an interval. For example, if a continuous function changes sign from positive to negative, then it must cross the x-axis somewhere in between, meaning there is at least one root. The theorem only works when continuity is present, which is why continuity over intervals matters so much in calculus.