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The idea of continuity lies in many things we experience in our daily lives, for instance, the time it takes you to log into StudyPug and read this section. Limits and continuity are so related that we cannot only learn about one and ignore the other. We will learn about the relationship between these two concepts in this section.

A function is

A function $f$ is

$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$

In simple words, the graph of a continuous function has no break in it and can be drawn without lifting your pen from the paper.

- 1.
__Introduction to Continuity__a)Discuss CONTINUITY in everyday language: a function whose graph has no break in it.b)Discuss CONTINUITY in the context of Calculus:

A function $f$ is continuous at a number a, if: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$

Classify different types of discontinuities: removable VS. infinite VS. jump

Analyze: rational function with a hole: $f(x)=\frac{(x-1)(x-2)}{(x-2)}$

VS. rational function with an asymptote: $g(x)=\frac{1}{x+1}$

VS. piecewise function:c)Lesson Overview - Continuity - 2.
**Discussing “Continuity” Graphically**

The graph of a function $f$ is shown below.a)State the numbers at which $f$ is discontinuous.b)Explain and classify each discontinuity.

- 3.
**Detecting Discontinuities**Are the following functions continuous at $x=3$?

i) $f(x)=\frac{x^2-2x-3}{x-3}$

ii)

iii) - 4.
**Discontinuities of Rational Functions (denominator=0)**

Locate and classify each discontinuity of the function: $f(x)=\frac{x^2-9}{x^2+x-6}$ - 5.
**Discussing “Continuity” Algebraically**

Find the values of a and b that make the function $f$ continuous on ($-\infty$, $\infty$).

1.

Limits

1.1

Finding limits from graphs

1.2

Limit laws

1.3

Continuity

1.4

Finding limits algebraically - direct substitution

1.5

Finding limits algebraically - when direct substitution is not possible

1.6

Infinite limits - vertical asymptotes

1.7

Limits at infinity - horizontal asymptotes

1.8

Intermediate value theorem

1.9

Squeeze theorem

We have over 170 practice questions in Differential Calculus for you to master.

Get Started Now1.1

Finding limits from graphs

1.3

Continuity

1.4

Finding limits algebraically - direct substitution

1.5

Finding limits algebraically - when direct substitution is not possible

1.6

Infinite limits - vertical asymptotes

1.7

Limits at infinity - horizontal asymptotes

1.8

Intermediate value theorem

1.9

Squeeze theorem