2026 FallSection 2.5 Continuity
Watch on YouTube →
Overview
Continuity at a point requires the function value to exist, the two-sided limit to exist, and the limit to equal the function value. The lesson applies this test to graphs and piecewise functions, distinguishes removable, jump, and infinite discontinuities, and uses factoring and common-denominator algebra to evaluate limits and repair removable holes.
Key takeaways
- Continuity at x = a requires three facts: f(a) is defined, the two-sided limit exists, and the limit equals f(a).
- A two-sided limit exists only when the left-hand and right-hand limits agree, as the unequal values -1 and 2 at x = 3 demonstrate.
- A removable discontinuity can be fixed by defining the function at the missing or mismatched point to equal the limit, such as setting h(3) = 6.
- For a piecewise function, check boundary points separately; at x = -2, g(-2) equals the right-hand limit but not the left-hand limit, making g right-continuous but not continuous.
- Jump and infinite discontinuities cannot be repaired by changing a function value at a single point because their one-sided behavior remains incompatible or unbounded.
- When direct substitution produces 0/0, use algebraic simplification, such as factoring or multiplying by a common denominator, before evaluating the limit.
Chapters
- Reviews limits of sequences and functions, including finite-point limits, limits at infinity, and one-sided limits.
- A function is continuous at x = a when f(a) exists, the limit as x approaches a exists, and that limit equals f(a).
- A function is continuous over an interval when it is continuous at every point in that interval; continuous functions also allow direct substitution to evaluate limits.
- At x = 1, the graph approaches a hole with y-coordinate 1 from both sides, but f(1) is undefined.
- At x = 3, the left-hand limit is -1 and the right-hand limit is 2, so the two-sided limit does not exist.
- At x = 5, both sides approach 1 while the solid point gives f(5) = 3, so the limit and function value differ.
- For the piecewise function with branches (1/2)x + 3 for x < -2 and x - 1 for x ≥ -2, only the boundary x = -2 needs checking because each branch is linear on its interval.
- The defined value is g(-2) = -3; the left-hand limit is 2 and the right-hand limit is -3.
- Since the one-sided limits differ, g is discontinuous at -2, but it is continuous from the right because g(-2) equals the right-hand limit.
- A shared factor such as x - 2 in the numerator and denominator cancels algebraically but leaves a hole; for (x² - x - 2)/(x - 2), the limiting value at 2 is 3.
- That hole is removable: defining the function's value at x = 2 as 3 makes the function continuous there.
- A vertical asymptote produces an infinite discontinuity, while unequal one-sided limits produce a jump discontinuity; neither is fixed by changing the value at one point.
- For h(x) = (x² - 9)/(x - 3) when x ≠ 3 and h(3) = 7, factoring gives a limit of 6, so redefining h(3) as 6 repairs the removable discontinuity.
- Direct substitution evaluates one example at x = 6 as 1/2; when substitution gives 0/0, algebra is needed instead.
- For the complex fraction limit as x approaches 3, multiplying by the common denominator 3x reduces the expression to 1/(3x), giving a limit of 1/9.
- The final graph has jump discontinuities at x = 0 and x = 1, and an infinite discontinuity at x = 3.
- The holes at x = -2, x = -0.5, and x = 5 are removable; set the function values to 2, 3.5, and 5 respectively to make the graph continuous at those points.
Transcript excerpt
0:10 Good morning class. Here's chapter 2 limits section 2.5 continued. So we've covered limits of sequences, limits of functions at infinity, limits of functions at finite numbers, algebraic method, and now continuity. Reminding you again of the same thing that was discussed, finding limits, numerical and graphical approach. Limit of f of x as x approaches l. We read the limit of f of x as x approaches a equ= limit of a7 as n approaches infinity is
1:00 l. We say a7 converges to l. Limit of a subn as n approaches infinity is infinity. Negative infinity or gn does not exist. We say the sequence diverges. To find limits simply plug in and observe. We are looking at three different graphs. All of which we are interested in finding the limit of f of x as x approaches a. When we approach a from the left, we are moving this way. We are moving up. As we approach it from the right, we are moving down here and here. And all of them takes us to this point with a y-coordinate of l. So the limit of f ofx as x approaches a does exist and it's equal to l.
1:54 What's the difference? F of A is L. F of A is B. F of A doesn't exist here. It's not defined. It's not given. Okay. Horizontal asmtote or h a if the limit of f of x as x approaches negative or positive infinity is l y= l is the equation of asmtote. If the limit of f ofx as x approaches a or a from the left or right is negative or positive infinity then the vertical asmtote va as the equation x= a and of course one-sided limits a function of the limit at a if the limit of f of x as x
This is the start of the transcript. Save this video free for the whole transcript, searchable and time-coded.
Related notes
Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Ara Araeipour.