Types of Continuity and Discontinuity Math Example 2

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Example 2

medium
Determine whether the piecewise function f(x)={x2x<13โˆ’xxโ‰ฅ1f(x) = \begin{cases} x^2 & x < 1 \\ 3 - x & x \geq 1 \end{cases} is continuous at x=1x = 1.

Solution

  1. 1
    Check f(1)=3โˆ’1=2f(1) = 3 - 1 = 2 (defined).
  2. 2
    Left-hand limit: limโกxโ†’1โˆ’x2=1\lim_{x\to 1^-} x^2 = 1.
  3. 3
    Right-hand limit: limโกxโ†’1+(3โˆ’x)=2\lim_{x\to 1^+}(3-x) = 2.
  4. 4
    Since limโกxโ†’1โˆ’f=1โ‰ 2=limโกxโ†’1+f\lim_{x\to1^-}f = 1 \neq 2 = \lim_{x\to1^+}f, the two-sided limit does not exist.
  5. 5
    This is a jump discontinuity at x=1x = 1.

Answer

Jump discontinuity at x=1x = 1 (left limit =1= 1, right limit =2= 2).
For piecewise functions, check the three continuity conditions. A jump discontinuity occurs when both one-sided limits exist but differ.

About Types of Continuity and Discontinuity

Continuity types classify how a function can fail to be continuous at a point. A removable discontinuity (hole) occurs when the limit exists but doesn't equal f(a). A jump discontinuity occurs when left and right limits differ. An infinite discontinuity occurs when the function approaches ยฑโˆž.

Learn more about Types of Continuity and Discontinuity โ†’

More Types of Continuity and Discontinuity Examples

Example 1 easy

Classify the discontinuity of [formula] at [formula].

Example 3 easy

Classify the discontinuity of [formula] at [formula].

Example 4 medium

Find the value of [formula] that makes [formula] continuous at [formula].

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