Can a function be continuous everywhere?

📌 CONCEPT: A function can be continuous everywhere, but it requires the function to be defined and have the same value at every point in its domain, with the limit of the function as x approaches a point being equal to the value of the function at that point.

📐 RULE / FORMULA: For a function f(x) to be continuous at a point x = a, the function must satisfy the following conditions: (1) f(a) is defined, (2) the limit of f(x) as x approaches a exists, and (3) the limit of f(x) as x approaches a equals f(a).

💡 WORKED EXAMPLE: Consider the function f(x) = |x| at x = 0. To check continuity, we must verify that f(0) is defined, the limit of f(x) as x approaches 0 exists, and this limit equals f(0). We find that f(0) = 0, the limit of f(x) as x approaches 0 is 0, and this limit equals f(0), so f(x) = |x| is continuous at x = 0.

⚠️ COMMON MISTAKE: Students often assume that a function is continuous everywhere simply because it has no discontinuities, but they fail to check if the function has any asymptotes or infinite limits, which can render the function discontinuous at those points.