Question

find the numbers at which f is discontinuous and determine whether f is continuous from the right, or from the left, or neither.

Answer 1

Where the function f is discontinuous  and continuous  is  mathematically given as

• f(x) is not continuous at x = 0.
• f(x) is left continuous at x = 1
• f(x) is right continuous for all x =5

What are the numbers at which f is discontinuous or continuous?

Generally, the equation for is  mathematically given as

Since x+1, 1/x, and (x-5), the function does not "break" since these three terms are continuous. at

• x = 1
• x = 5.

According to the dictionary definition of continuity, the function f(x) is continuous at x = a if:

lim (x->a-) f(x) = lim (x->a+) f(x) = f(a).

for x = 0

lim (x->0-) f(x)

= lim (x->0-) (x+1),

since f(x) = x+1 for $x \leq 1$

f(x<=1) = 1 + 0

f(x<=1)= 1

lim (x->0+) f(x)= lim (x-->0+) ( √(x-5) ), since f(x)

lim (x->0+) f(x)= √(x-5 for x>=5

lim (x->0+) f(x)= 5

In conclusion,

• f(x) is not continuous at x = 0.
• f(x) is left continuous at x = 1
• f(x) is right continuous for all x =5

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Complete Question

Find the numbers at which f is discontinuous. Then determine whether f is continuous from the right, from the left, or neither at each point of discontinuity.

f(x)={ x+1 if x< 1

1/x if 1<x<5

√(x-5) if x> 5