Question

Consider a function f(x) defined as follows:

f(x) = 1/x for x ≠ 0 and f(0) = 0.

Prove that the function f(x) is continuous at every point except x = 0.



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Answer 1

o prove that f(x) is continuous at every point except x = 0, we need to show that:

f(x) is defined at every point except x = 0

lim(x→a) f(x) exists for every a ≠ 0

lim(x→0) f(x) exists

First, we note that f(x) is defined for all values of x except x = 0 because we have defined it as such.

Next, we consider the limit as x approaches a for a ≠ 0:

lim(x→a) f(x) = lim(x→a) 1/x = 1/a

Since the limit exists and is finite for all values of a ≠ 0, we conclude that f(x) is continuous at every point except x = 0.

Finally, we consider the limit as x approaches 0:

lim(x→0) f(x) = lim(x→0) 1/x

This limit does not exist because the function approaches positive infinity as x approaches 0 from the right, and negative infinity as x approaches 0 from the left. Therefore, we conclude that f(x) is not continuous at x = 0.

Hence, we have shown that f(x) is continuous at every point except x = 0

Explanation: