2.5 Defn: f is continuous at a if lim_{x→a}f(x) = f(a) (i.e., if lim_{x→a}f(x) = f(lim_{x→a}x) Examples: Read left and right continuity If f, g continuous at a, c ∈ R, then f + g, fg, cf, f/g (if g(a) ≠ 0) are continuous. If g continuous at a and f continuous at g(a), then f ∘ g continuous at a. Ex: lim_{x→0} \frac{x^{2}−e^{x^{3}}}{cos(x)} = Intermediate value theorem: Suppose f continuous on [a, b], f(a) ≠ f(b) and n is between f(a) and f(b), then there exists c ∈ (a, b) such that f(c) = N. Example: Show that x^{2} − 7x + 1 has a root between 0 and 1.