Continuity When Will It End

For functions that are "normal" enough, we know immediately whether or not they are continuous at a given point. Nevertheless, the continuity of a function is such an important property that we need a precise definition of continuity at a point: A function f (x) is continuous at c if and only if

In other words if as you get closer to c from both sides there is a value you are getting closer to and it is the value of the function at c. Or If when you trace over a point and you don’t have to lift your pencil to draw the graph, the graph is continuous at that point

1. If f is continuous at every real number c, then f is said to be continuous.

2. If f is not continuous at c, then f is said to be discontinuous at c. The function f can be discontinuous for two distinct reasons: a) f(x) does not have a limit as x approaches c The function is discontinuous at x=0 The function is discontinuous at x=1

b) The value of the function at c does not equal the limit or the function is undefined at that value The function is discontinuous at x=1

Endpoint: A function y=f(x) is continuous at a left endpoint or is continuous at a right endpoint b of its domain if The graph is continuous at x = 3 The graph is not continuous at x = -3

There are different types of discontinuity Jump Discontinuity Infinite Discontinuity Oscillating Discontinuity Removable Discontinuity

In both examples the function would be continuous if the point (1,-1) wasn’t removed

Jump Discontinuity Jump discontinuity is when the left and right handed limits have different values

Infinite Discontinuity Infinite discontinuity happens when there is a vertical asymptote.

This is the equation of y=sin(1/x) it oscillates to much to have a limit as therefore it has oscillating discontinuity at x=0 Oscillating Discontinuity

Find the points and the type of discontinuity Look for asymptotes or holes, in other words values the function is undefined. In piecewise functions look for where the graph jumps or there are holes.

This graph has infinite discontinuity at x=5.

This graph has a hole, therefore at x= -2 there is removable discontinuity.

The graph has jump discontinuity at x=1 and removable discontinuity at x=3