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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
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
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