Continuity & Discontinuity

Unit 1 · Limits & Continuity

Continuity at a Point

A function f is continuous at x = c if three conditions all hold: f(c) is defined, the two-sided limit exists, and the limit equals f(c). Failing any one condition produces a discontinuity. Removable discontinuities can be fixed by redefining f(c).

Three Conditions for Continuity at x = c

(i)

f(c) \text{ is defined}

(ii)

\lim_{x \to c} f(x) \text{ exists}

(iii)

\lim_{x \to c} f(x) = f(c)

Types of Discontinuity

Removable (hole)

\lim_{x\to c} f(x) \text{ exists, but } f(c) \text{ undefined or } \neq L

Jump

\lim_{x\to c^-} f(x) \neq \lim_{x\to c^+} f(x)

Infinite

\lim_{x\to c} f(x) = \pm\infty

AP Tip: Polynomials are continuous everywhere. Rational functions are continuous wherever the denominator ≠ 0.

Type 1

Checking Continuity — Three Conditions

Verify each of the three conditions in order. If any fails, identify the type of discontinuity.

Example 1

Is f continuous at x = 3? Identify which condition(s) fail.

f(x) = \begin{cases} 2x + 1 & x < 3 \\ 10 & x = 3 \\ x^2 - 2 & x > 3 \end{cases}

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Type 2

Removing a Discontinuity

Find the value k that makes f continuous: set k equal to the limit at the point of discontinuity.

Example 2

Find k that makes g continuous at x = 5.

g(x) = \begin{cases} \dfrac{x^2 - 3x - 10}{x - 5} & x \neq 5 \\ k & x = 5 \end{cases}

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Type 3

Finding k for Continuity

For piecewise functions, set the left-hand and right-hand limits equal at the junction point and solve for the unknown constant.

Example 3

Find k so that f is continuous everywhere.

f(x) = \begin{cases} kx + 3 & x \leq 2 \\ x^2 + 1 & x > 2 \end{cases}

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