Continuity vs. Differentiability

Differentiable Function

A continuous function where a derivative can be found at every point within its domain

Cusp (Non-differentiable)

Basically a rounded corner function; ex: F(x)=x^2/3 there is a critical point but no

Continuity

Being able to draw the graph of a function without having to lift up a pencil basically

Jump Discontinuity

Common in a piece-wise function where limx->a from the left is not equal to that of the limit coming from the right.

Corner (Non-differentiable)

Ex: Absolute Value Functions

Definition of Derivative

Exists for each side of limit for a point in a differentiable function

Continuous function

Function is continuous at every point in its domain (does not mean it has to be continuous on every interval)

Intermediate Value Theorem

If f(x) is continuous on some interval [a,b] and some y value is between f(a) and f(b), then f(c)=y for the value c which is an x value on the interval [a,b]

Oscillating Discontinuity

Typical with a sinusoidal graph, this is where as the graph approaches zero, the values are no longer defined but the graph is limited between 1 and -1, ish

Vertical Tangent (Non-differentiable)

When a line with an undefined slope (undefined derivative) can be intersect the function at a single point (tangent)

Removable Discontinuity

When one value can be added to the graph in order to make it continuous again

Infinite Discontinuity

Where the limit of a function approaching an x-value is infinity and/or -infinity from either side of the x-value

Continuity at a point

limit x->c of f(x) = f(c)