What does continuous at a point mean?

Answer 1
The definition for continuity at a point #a# is #lim_(x->a-)f(x)=f(a)=lim_(x->a+)f(x)#.

The simplest explanation is that you must draw a curve through the point without lifting your pen. Lifting your pen would be a discontinuity.

Continuity at a point allows us to define and come up with theorems about continuous functions. The continuous functions (in their domain) are: polynomials, rationals, roots, trigonometric, exponential, and logarithmic. Be careful because these functions are not continuous for all reals; it is important to note that there is a restriction "in their domain".

For calculus, you will notice most theorems require continuity, such as the IVT, MVT, derivative tests, and FTC to name a few.

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Answer 2

Continuous at a point means that a function is defined at that point and the limit of the function as the input approaches that point exists and is equal to the value of the function at that point.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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