Question

What does continuous at a point mean?

Answer 1

The definition for continuity at a point `a` is `lim_(x->a-)f(x)=f(a)=lim_(x->a+)f(x)`.

The simplest explanation is that you must draw a curve through the point without lifting your pen. Lifting your pen would be a discontinuity.

Continuity at a point allows us to define and come up with theorems about continuous functions. The continuous functions (in their domain) are: polynomials, rationals, roots, trigonometric, exponential, and logarithmic. Be careful because these functions are not continuous for all reals; it is important to note that there is a restriction "in their domain".

For calculus, you will notice most theorems require continuity, such as the IVT, MVT, derivative tests, and FTC to name a few.