Continuity In Functions: Definition And Conditions Explained

f is continuous at x=c if…

If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k

A function f is said to be continuous at x = c if the following three conditions are satisfied:

1. f(c) is defined,
2. The limit of f(x) as x approaches c exists, and
3. The limit of f(x) as x approaches c is equal to f(c).

The first condition requires that the function has a defined output at the point where x equals c. This means that the function is well-behaved and does not have any undefined values at this point.

The second condition involves finding the limit of the function as x approaches c. This means that we need to look at the behavior of the function as x gets very close to c and see what value it appears to be approaching. If the limit exists, this means that the function is approaching a fixed value as we get closer and closer to c.

Finally, the third condition states that the limit value for the function as x approaches c must be equal to f(c). In other words, there cannot be any jumps or discontinuities in the function at x=c. The function must approach the same value from both the left and the right sides, and this value must be equal to f(c).

If all three of these conditions are met, we can say that f is continuous at x=c.

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