Understanding Continuous Functions and the Intermediate Value Theorem in Calculus

What does it mean when a function is continuous?

Means we can draw the graph without lifting pencil from paper; no holes, breaks or jumps on intervalf(c) exists in the domain of flimit as x approaches c of f(x) exists

A function is said to be continuous at a point if it is defined at that point, its limit exists at that point, and the limit is the same as the function value. In simpler terms, a function is continuous if it can be drawn without lifting the pen from the paper.

If a function is said to be continuous on an interval, that means the function is continuous at every point within that interval. Continuous functions have many important properties, including the intermediate value theorem, which states that if a continuous function takes on two values, then it must also take on every value between those two values at some point within the interval. This is a key property that is used in solving many calculus problems.

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