Is a function continuous at a point discontinuity?

A function being continuous at a point means that the two-sided limit at that point exists and is equal to the function’s value. Point/removable discontinuity is when the two-sided limit exists, but isn’t equal to the function’s value.

Can a function be continuous and differentiable at the same point?

We see that if a function is differentiable at a point, then it must be continuous at that point. There are connections between continuity and differentiability. If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .

What if LHL is not equal to RHL?

Originally Answered: If LHL and RHL is finite but not equal then does limt exists? No. Yes; it’s called a double-sided limit, and each individual limit is a one-sided limit. See One-sided limit – Wikipedia for more.

What makes a function not continuous?

In other words, a function is continuous if its graph has no holes or breaks in it. For many functions it’s easy to determine where it won’t be continuous. Functions won’t be continuous where we have things like division by zero or logarithms of zero.

Why should a function be continuous and differentiable?

Hence, differentiability is when the slope of the tangent line equals the limit of the function at a given point. This directly suggests that for a function to be differentiable, it must be continuous, and its derivative must be continuous as well. Thus, a differentiable function is also a continuous function.

Why is a differentiable function always continuous?

In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. Informally, this means that differentiable functions are very atypical among continuous functions.

What makes a graph continuous?

What are Continuous Graphs? Continuous graphs are graphs where there is a value of y for every single value of x, and each point is immediately next to the point on either side of it so that the line of the graph is uninterrupted. In other words, if the line is continuous, the graph is continuous.