To understand continuity, Sagemath is the best tool to demonstrate it. Just see the plots of piecewise functions and you will come to know whether it is continuous or not.

**Continuous Functions :-** A function is said to be **continuous** on an interval when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks. If some function f(x) satisfies these criteria from x=a to x=b, for example, we say that f(x) is continuous on the interval [a, b]. The brackets mean that the interval is **closed** — that it includes the endpoints a and b. In other words, that the interval is defined as a ≤ x ≤ b. An **open** interval (a, b), on the other hand, would not include endpoints a and b, and would be defined as a < x < b.

To think of it another way, if you can trace a function on an interval without picking up your pen (and without running over any holes), the function is continuous on that interval.

1) Plot x^{2} from x=0 to x=1

2) Plot -x+2 from x=1 to x=2

3) Plot x^{2}-3*x+2 from x=2 to x=3

4) Create a black point at (0, 0)

5) Create another black point at (3, 2)

6) Combine the plots and points into a single graph with the given bounds

Sage Code to plot it is:

p1 = plot(x^2, x, 0, 1) p2 = plot(-x+2, x, 1, 2) p3 = plot(x^2-3*x+2, x, 2, 3) pt1 = point((0, 0), rgbcolor='black', pointsize=30) pt2 = point((3, 2), rgbcolor='black', pointsize=30) (p1+p2+p3+pt1+pt2).show(xmin=0, xmax=3, ymin=0, ymax=2)

Output is:

As you can see, the function travels from x=0 to x=3 without interruption, and since the two endpoints are closed (designated by the filled-in black circles), f(x) is continuous on the closed interval [0, 3].

1) Plot 2*x from x=0 to x=1

2) Plot -x+3 from x=1 to x=2

3) Plot -(x-3)+2 from x=2 to x=3

4) Create a dashed line to indicate that the function jumps to a y-value of 3 when x is equal to 1

5-10) Create open (faceted) or closed (filled-in) points to indicate whether the intervals are open or closed

11) Combine the plots, line, and points into a graph with the given bounds

Sage code is:

p1 = plot(2*x, x, 0, 1) p2 = plot(-x+3, x, 1, 2) p3 = plot(-(x-3)^3+2, x, 2, 3) l1 = line([(1, 2.1), (1, 2.9)], linestyle='--') pt1 = point((0, 0), rgbcolor='black', pointsize=30) pt2 = point((1, 2), rgbcolor='white', faceted=True, pointsize=30) pt3 = point((1, 3), rgbcolor='black', pointsize=30) pt4 = point((2, 1), rgbcolor='black', pointsize=30) pt5 = point((2, 3), rgbcolor='white', faceted=True, pointsize=30) pt6 = point((3, 2), rgbcolor='black', pointsize=30) (p1+p2+p3+l1+pt1+pt2+pt3+pt4+pt5+pt6).show(xmin=0, xmax=3, ymin=0, ymax=3)

The function shown in the graph is *not* continuous on the closed interval [0, 3], since it has **discontinuities** at both x=1 and x=2. A discontinuity is any x-value at which a function has an interruption, break or jump — something that would require you to pick up your pen if you were tracing the function. The filled-in black circles, again, indicate that the interval includes that point, while the open circles indicate that the interval excludes that point. The dashed line at x=1 shows that f(1) = 3.

Sage Code:

# exp() means 'e to the ___' p1 = plot(exp(x/5)-1, x, 0, e) p2 = plot(ln(x), x, e, e^2) pt1 = point((0, 0), rgbcolor='black', pointsize=30) pt2 = point((e, exp(e/5)-1), rgbcolor='black', pointsize=30) pt3 = point((e, 1), faceted=True, rgbcolor='white', pointsize=30) pt4 = point((e^2, 2), rgbcolor='black', pointsize=30) (p1+p2+pt1+pt2+pt3+pt4).show(ymin=0, ymax=2)

The function is not continuous on the closed interval [0, e²], as it has a discontinuity at x=e

# exp() means 'e to the ___' p1 = plot(exp(x/5)-1, x, 0, e) p2 = plot(ln(x), x, e, e^2) pt1 = point((0, 0), rgbcolor='black', pointsize=30) pt2 = point((e, exp(e/5)-1), rgbcolor='black', pointsize=30) pt3 = point((e, 1), faceted=True, rgbcolor='white', pointsize=30) pt4 = point((e^2, 2), rgbcolor='black', pointsize=30) (p1+p2+pt1+pt2+pt3+pt4).show(ymin=0, ymax=2)