Quick Answer: Is A Function Differentiable If It Is Continuous?

Why does a function have to be continuous to be differentiable?

Until then, intuitively, a function is continuous if its graph has no breaks, and differentiable if its graph has no corners and no breaks.

So differentiability is stronger.

A function is only differentiable on an open set, then it has no sense to say that your function is differentiable en a or on b..

Where is a function continuous but not differentiable?

The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable.

How do you know if a function is not differentiable?

We can say that f is not differentiable for any value of x where a tangent cannot ‘exist’ or the tangent exists but is vertical (vertical line has undefined slope, hence undefined derivative).

How do you know if a function is continuous without graphing?

How to Determine Whether a Function Is Continuousf(c) must be defined. The function must exist at an x value (c), which means you can’t have a hole in the function (such as a 0 in the denominator).The limit of the function as x approaches the value c must exist. … The function’s value at c and the limit as x approaches c must be the same.

How do you know if a function is differentiable?

If a function f(x) is differentiable, then f'(x) may or may not be continuous, let alone differentiable. If f'(x) does happen to be continuous, we say f(x) is continuously differentiable. f(x) is differentiable everywhere, but f'(x)=|x|, which is continuous but not differentiable at 0.

Can a piecewise function be continuous?

The piecewise function f(x) is continuous at such a point if and only of the left- and right-hand limits of the pieces agree and are equal to the value of the f. …

What kinds of functions are not differentiable?

Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. There are however stranger things. The function sin(1/x), for example is singular at x = 0 even though it always lies between -1 and 1.

Where is the function 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.

How do you tell if a function is continuous or not?

If a function f is only defined over a closed interval [c,d] then we say the function is continuous at c if limit(x->c+, f(x)) = f(c). Similarly, we say the function f is continuous at d if limit(x->d-, f(x))= f(d).