Continuity and One-sided Limits. A function f is continuous at x = c if there is no interruption in the graph at c The graph is unbroken at c , with no holes, jumps, or gaps A function is continuous at c if f(c) is defined lim f(x) exists lim f(x) = f(c). x c. x c.
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Theorem 1.10 Existence of a Limit
If f is a function and c and L are real numbers, then lim f(x) = L if and only if lim f(x) = L and lim f(x) = L
Theorem 1.11 Properties of Continuity
If b is a real number and f and g are functions that are continuous at c, then the following functions are also continuous at c
Theorem 1.12 Continuity of Composite Functions
If g is continuous at c and f is continuous at g(c), then the composite function f◦g(x) = f(g(x)) is continuous at c
Theorem 1.13 Intermediate Value Theorem
If f is continuous on the closed interval [a, b], f(a) ≠ f(b), and k is any real number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k