Calculus

Continuity

Continuity Warmup

         

Which of the graphs show a function that is continous at x=a?x=a?

g(x)={1xif x0aif x=0g(x) = \begin{cases} \frac{1}{x} & \textrm{if } x \neq 0 \\ a & \textrm{if } x = 0 \\ \end{cases}

Is there a real number aa for which the function g(x)g(x) is continuous at x=0?x = 0?

f(x)=xx,g(x)=x2x,h(x)=xx2f(x)=\frac { x }{ x } ,g(x)=\frac { x^{ 2 } }{ x } ,h(x)=\frac { x }{ x^{ 2 } }

Each of these functions is undefined at x=0.x = 0. Which functions is it possible to extend by defining a functional value at x=0x = 0 in such a way that the resulting extended function is continuous at x=0?x = 0?

True or False?

The function f(x)=x3+x1f(x) = x^3 + x - 1 has a root between 0 and 1.

Hint: f(0)=1f(0) = -1 and f(1)=1.f(1) = 1. Apply the Intermediate Value Theorem.

Which of the graphs shows a function f(x)f(x) for which limxaf(x)\lim_{x\to a} f(x) exists?

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