Calculus 1000- Continuity (Sec 007, October 5, 2026)

Overview

Asghar Ghorbanpour reviews continuity as equality between a function’s value and its limit, then shows how continuity is preserved by sums, products, quotients, and composition when domain conditions are satisfied. For a piecewise function, he finds the boundary conditions a = 0 and b = -3, emphasizes proving continuity on every interval—not just at the breakpoints—and previews the Intermediate Value Theorem for the next class.

Key takeaways

Continuity at x = a requires all three conditions together: f(a) is defined, lim(x→a) f(x) exists, and that limit equals f(a).
For the piecewise example, continuity at x = −1 equates the left limit a − 2 with the rational-piece limit and function value −2, yielding a = 0.
At x = 0, the rational piece approaches −3 while b sin x/x approaches b because lim(x→0) sin x/x = 1; continuity therefore gives b = −3.
A proof of continuity on the whole real line must establish continuity within every piece’s interval as well as at the formula-change points.
Continuity is preserved by composition when the inner function is continuous at the input point and the outer function is continuous at the inner function’s output.
In the worked composition limit, f(x) approaches −2 and x² approaches 1 as x approaches 1, so sin(f(x) + x²) approaches sin(−1).

Chapters

0:00 Continuity at a Point: Matching the Function Value to Its Limit

A function is continuous at x = a when its limit as x approaches a exists and equals f(a).
Unlike earlier limit calculations, continuity requires checking the function’s actual value at the point.

3:00 Polynomials, Rational Functions, and Trigonometric Continuity

Direct substitution shows every polynomial is continuous at every real number.
Rational functions are continuous wherever their denominator is nonzero, which is precisely the restriction imposed by their domains.
The graphs of sin x and cos x illustrate that both trigonometric functions are continuous across the real line.

7:30 Limit Laws Preserve Continuity Under Algebraic Operations

If f and g are continuous at x = a, then f + g, f − g, cf, and fg are continuous there.
The quotient f/g is continuous at a when g(a) is nonzero; roots and other operations also require their domain conditions to hold.
These continuity rules follow from the corresponding limit laws and let students build new continuous functions from known ones.

11:20 Finding the Breakpoints in a Piecewise Continuity Problem

The example asks for parameters a and b that make a piecewise-defined function continuous on the entire real line.
Ghorbanpour identifies x = −1 and x = 0 as the formula-change points requiring separate continuity checks.
At each breakpoint, the one-sided limits must agree with each other and with the function’s assigned value.

16:00 At x = −1, Match the Linear and Rational Pieces to Find a

Approaching −1 from the left uses the linear expression 2x + a, giving a left-hand limit of a − 2.
Approaching from the right uses the rational piece (x² + 3)/(x − 1), whose limit at −1 is −2.
The function’s value at −1 is also −2, so continuity requires a − 2 = −2 and therefore a = 0.

24:00 At x = 0, Use the Sine Limit to Find b

The rational piece has limit (0² + 3)/(0 − 1) = −3 as x approaches 0 from the left, matching the function value at 0.
The right-hand piece contains b sin x/x; using the standard limit lim(x→0) sin x/x = 1 gives a right-hand limit of b.
Continuity at zero requires b = −3; Ghorbanpour notes that students should cite the established sine limit in a written solution.

29:37 Complete the Proof by Checking Continuity on Every Piece

Checking only x = −1 and x = 0 is insufficient when the question asks for continuity on all of ℝ.
On the interval x < −1, the function is polynomial, so it is continuous throughout that interval.
Between −1 and 0, the rational expression is defined and continuous; for x > 0, b sin x/x is a quotient of continuous functions with x ≠ 0.
Ghorbanpour says omitting these interval checks cost many students roughly one-third of the available grade on a midterm problem.

39:05 The Composition Limit Rule Requires Continuity at the Inner Limit

If lim(x→a) g(x) = L and f is continuous at L, then lim(x→a) f(g(x)) = f(L).
The continuity condition must hold at the value L approached by the inner function, not merely at an unrelated point.
This rule explains when a limit can be passed through an outer function.

41:24 Apply Composition to a Trigonometric Limit and Preview the IVT

For the example with lim(x→1) f(x) = −2, the inner expression f(x) + x² approaches −2 + 1 = −1, so continuity of sine gives the limit sin(−1).
If g is continuous at a and f is continuous at g(a), then the composition f∘g is continuous at a, subject to domain restrictions.
Ghorbanpour closes by previewing the Intermediate Value Theorem, which will show how to prove that certain values are attained without explicitly finding where.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Asghar Ghorbanpour.