Question

Using the sequential definition of continuity, prove that the function f: R → R ,
defined by f(x) = 3x^2+ 7 ,∀x ∈R,
is continuous.

Answer 1

Let x0 be any real number. Let's prove that f(x) is continuous in x0. Sequential definition of continuity states that function is continuous in a dot x0 if for every x_n sequence $lim_{n\to \infty}f(x_n)=f(x_0)$ .

Indeed, if $x_n\to x_0$ then $f(x_n)=3\cdot x_n^2+7\to3\cdot x_0^2+7=f(x_0)$, which proves continuity in x0.