Question

In calculus, we define a function f to be continuous at a real number a provided that for every " > 0, there exists a ı > 0 such that if jx aj < ı, thenjf .x/f .a/j<".

Note: The symbol " is the lowercase Greek letter epsilon, and the symbol ı is the lowercase Greek letter delta.
Complete each of the following sentences using the appropriate symbols for quantifiers:
(a) Afunctionf iscontinuousattherealnumberaprovidedthat::::
(b) Afunctionf isnotcontinuousattherealnumberaprovidedthat::::
Complete the following sentence in English without using symbols for quan- tifiers:
(c) Afunctionf isnotcontinuousattherealnumberaprovidedthat::::

Answer 1

In calculus, the definition of a function being continuous at a real number a is that for every ε > 0, there exists a δ > 0 such that if |x - a| < δ, then |f(x) - f(a)| < ε. A function f is continuous at a provided that this condition holds, while it is not continuous at a if there exists an ε > 0 such that for any δ > 0, there exists an x such that |x - a| < δ and |f(x) - f(a)| ≥ ε.

Step-by-step explanation:

In calculus, we define a function f to be continuous at a real number a provided that for every ε > 0, there exists a δ > 0 such that if |x - a| < δ, then |f(x) - f(a)| < ε.

(a) A function f is continuous at the real number a provided that for every ε > 0, there exists a δ > 0 such that if |x - a| < δ, then |f(x) - f(a)| < ε.

(b) A function f is not continuous at the real number a provided that there exists an ε > 0 such that for every δ > 0, there exists an x such that |x - a| < δ and |f(x) - f(a)| ≥ ε.

(c) A function f is not continuous at the real number a provided that there exists an ε > 0 such that for any δ > 0, there exists an x such that |x - a| < δ and |f(x) - f(a)| ≥ ε.

Answer 2

A function is continous if there exists
`a` such that
`\lim_(x\to a) f(x)=f(a)`.

Hope this helps.