Continuity At A Point

When is a function continuous?

If you can draw the graph of the function without lifting your pencil, then it is continuous. In a continuous function, every point in the interval [a,b] must connected.

lim_x->cf(x) = f(c)

What are points of discontinuity?

- holes
- jumps
- asymptotes
- oscillations

The first three, more common, points of discontinuity are shown in the graph above.

Hole discontinuities can be fixed, or removable, by filling them in with piecewise equations.

How To Remove Discontinuities:

1. factor the equation
2. the x value that cancels out in the numerator and denominator is the hole
3. take that x value and substitue it in to the now factored and reduced f(x)
4. set up the piecewise equation for the hole

EXAMPLE

Problem: f(x) = [(x²+2x+1)/(x²-3x-4)]

Solution: [(x²+2x+1)/(x²-3x-4)] = [(x+1)(x+1)]/[(x-4)(x+1)] = (x+1)/(x-4) , x=-1 is hole

y={ [(x²+2x+1)/(x²-3x-4)], x≠1
0, x=1