# Continuous and Discontinuous Functions

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### Continuous and Discontinuous Functions :

Continuous and Discontinuous Functions

### Continuous Functions :

Continuous Functions Consider the graph of f(x) = x3 − 6x2 − x + 30: We can see that there are no "gaps" in the curve. Any value of x will give us a corresponding value of y. We could continue the graph in the negative and positive directions, and we would never need to take the pencil off the paper. Such functions are called continuous functions.

### Functions With Discontinuities :

Functions With Discontinuities Now consider the function We note that the curve is not continuous at x = 1.

### Slide 4:

We observe that a small change in x near x = 1 gives a very large change in the value of the function. For a function to be continuous at a point, the function must exist at the point and any small change in x produces only a small change in f(x). In simple English: The graph of a continuous function can be drawn without lifting the pencil from the paper. Many functions have discontinuities

### Split Functions (Piecewise-defined functions) :

Split Functions (Piecewise-defined functions) Most functions you are familiar with are defined in the same manner for all values of x. However, there are some functions which are defined differently in different domains. These are known as split functions (or piecewise-defined functions).

### EX: f(x) = -x2 + 4 :

EX: f(x) = -x2 + 4 This function is not a split function. It is defined the same way for all values of x. To find the value of the function at a given x-value, simply substitute into f(x) = -x2 + 4

### Slide 7:

In the region x < 1, we have a straight line with slope 2 and y-intercept 3. As x approaches 1, the value of the function approaches 5 (but does not reach it because of the "<" sign). Now for the region x ≥ 1. When x = 1, the function has value f(-1) = -(1)2 + 2 = -1 +2 = 1.

### Slide 8:

As we go further to the right, the function takes values based on f(x) = -x2 + 2. It is a parabola This function has a discontinuity at x = 1, but it is actually defined for x = 1 (and has value 1).

### Even and Odd Functions :

Even and Odd Functions

### Even Functions :

Even Functions A function y = f(t) is said to be even if f(-t) = f(t) for all values of t. The graph of an even function is always symmetrical about the vertical axis (that is, we have a mirror image through the y-axis).

### Cosine curvef(t) = 2 cos πt :

Cosine curvef(t) = 2 cos πt Notice that we have a mirror image through the f(t) axis.

### Even Square wave: :

Even Square wave:

### Triangular wave: :

Triangular wave: In each case, we have a mirror image through the f(t) axis. Another way of saying this is that we have symmetry about the vertical axis.

### Odd Functions :

Odd Functions A function y = f(t) is said to be odd if f(-t) = - f(t) for all values of t. The graph of an odd function is always symmetrical about the origin.

### Examples of Odd Functions :

Examples of Odd Functions Sine Curve y(x) = sin x Notice that if we fold the curve along the y-axis, then along the t-axis, the graph maps onto itself. It has origin symmetry.

"Saw tooth" wave

### Odd Square wave :

Odd Square wave Each of these three curves is an odd function, and the graph demonstrates symmetry about the origin.

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